Newton Polygons of Zeta Functions and L-Functions

By DAQING WAN

§0. Introduction

In this paper, we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeros or poles of zeta functions and L-functions. Our main object is the Adolphson-Sperber conjecture [2] which asserts that under a simple condition the generic Newton polygon of L-functions coincides with its lower bound. We show that the conjecture is false in its full form but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(x_{1}, ⋯, x_{n}) of degree d_{i} with respect to x_{i} (1 ≤ i ≤ n), where the d_{i} are fixed positive integers. Then for all large prime number p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. Our main results are several decomposition theorems obtained using certain maximizing functions from linear programming. In particular, this suggests a possible connection between Newton polygons and resolution of singularities of toric varieties.

Let p be a prime, q = p^{a}, F_{q} the finite field of q elements, and F_{q^{m}} its extension of degree m. Fix a nontrivial additive character Ψ of F_{p}. For any Laurent polynomial f(x_{1}, ⋯, x_{n}) ∈ F_{q}[x_{1}, x_{1}^{−1}, ⋯, x_{n}, x_{n}^{−1}], we form the exponential sum

S_{m}^{∗}(f) = ∑ Ψ(Tr_{F_{q^{m}} /F_{p}}(f(x_{1}, ⋯, x_{n}))),  (0.1)

where the sum is over all (x_{1}, ⋯, x_{n}) ∈ (F_{q^{m}}^{∗})^{n}. By well-known theorems of Dwork-Bombieri-Grothendieck, the following L-function is a rational function:

L^{∗}(f, t) = exp( ∑_{m=1}^{∞} S_{m}^{∗}(f)\frac{t^{m}}{m} ) = \frac{∏_{i}(1 − α_{i}t)}{∏_{j}(1 − β_{j}t)}.  (0.2)

Equivalently,

S_{m}^{∗}(f) = β_{1}^{m} + β_{2}^{m} + ⋯ − α_{1}^{m} − α_{2}^{m} − ⋯,

where the α_{i} and β_{j} are algebraic integers. Thus, finding a sharp p-adic estimate for the sums S_{m}^{∗}(f) is reduced to determining the p-adic values of the reciprocal roots α_{i} and the reciprocal poles β_{j}. This question can best be described in terms of Newton polygons, see [11].

The Laurent polynomial f is a sum of monomials f = ∑_{j} a_{j}x^{V_{j}} and as such has a well defined Newton polyhedron Δ(f) at infinity. This is the convex closure in R^{n} generated by the origin and the lattice points V_{j}. Assuming that f is nondegenerate, Adolphson and Sperber [2] proved that L^{∗}(f, t)^{(−1)^{n−1}} is a polynomial of degree n!V(f), where V(f) is the volume of Δ(f). Furthermore [2], the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} lies above a certain explicit combinatorial lower bound P (Δ) involving lattice points in Δ(f) and its rational multiples rΔ(f). Adolphson and Sperber [2] conjectured that the Newton polygon coincides generically with its lower bound if p lies in a certain residue class.

To describe this conjecture more fully, we fix such an n-dimensional integral Newton polyhedron Δ in R^{n} containing the origin. For each face σ_{i} of codimension 1 not containing the origin, let ∑_{k=1}^{n} α_{ik}X_{k} = 1 be the equation of the unique hyperplane passing through σ_{i}. The α_{ik} are rational numbers. Define the denominator D(Δ) of Δ to be the least common denominator of all the α_{ik}. Let M_{p}(Δ) be the moduli space of all Laurent polynomials over F̄_{p} with Δ(f) contained in Δ. It is clear that M_{p}(Δ) is isomorphic to the affine space (F̄_{p})^{b} for a certain positive integer b. Hence, M_{p}(Δ) is an irreducible quasi-projective variety. Let H_{p}(Δ) be the moduli space of those f ∈ M_{p}(Δ) such that Δ(f) = Δ, f is nondegenerate with respect to Δ and the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides with the given lower bound P (Δ). In Dwork’s terminology, H_{p}(Δ) is called the Hasse domain of the generic Laurent polynomial f defined over F̄_{p} with Δ(f) contained in Δ. It is not difficult to prove that H_{p}(Δ) is a Zariski open subset of M_{p}(Δ) (possibly empty). Moreover (see the end of §2), the complement of H_{p}(Δ) in M_{p}(Δ) is an affine variety defined by a polynomial in the variables a_{j}, called the Hasse polynomial and denoted by h_{p}(Δ). A basic question is whether or not H_{p}(Δ) is Zariski dense in M_{p}(Δ), in other words, whether or not the Hasse polynomial h_{p}(Δ) is non-zero. Based on earlier results of Sperber [12-14] and Carpentier [4] about p-adic estimates of exponential sums, Adolphson and Sperber [2] made the following

Conjecture 1. For all large prime p satisfying p ≡ 1 (mod D(Δ)), the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides generically with its lower bound, i.e., for such p, H_{p}(Δ) is Zariski dense in the moduli space M_{p}(Δ).

It can be shown that Conjecture 1 generalizes a conjecture of Dwork [6, p.40] and Mazur [11, p.661] (for large p) which asserts that for a general hypersurface X of degree d, the Newton polygon of the interesting part of the zeta function Z(X, t) coincides with its lower bound (the Hodge polygon). Katz (unpublished, but see [11]) proved that the conjecture is true for the first slope. Miller (see [9]) and Koblitz [9] proved that the conjecture is true for the horizontal side. In his second paper on the Weil conjectures, Deligne [5] announced an ℓ-adic proof of the Dwork-Mazur conjecture. However, Deligne’s proof has not been published. Recently, Illusie [10] found a new proof using crystalline cohomology. Similar results on other special families of varieties (curves of genus g, polarized abelian varieties of dimension g, K_{3} surfaces) were obtained by Miller, Koblitz, Mumford, Norman-Oort and Ogus, see [10] and [11]. One consequence of this paper is a p-adic proof of the Dwork-Mazur conjecture.

In this paper, we shall prove

Theorem 2. Conjecture 1 is false for all n ≥ 5.

Our counterexample of Conjecture 1 is an n-dimensional simplex (n ≥ 5) on which there are no lattice points other than the vertices. Thus, the general Laurent polynomial in this case is simply the “general diagonal form”. One can then compute the L-function explicitly and use Stickelberger’s theorem.

One of the main results of this paper is

Theorem 3. There is an integer D^{∗}(Δ) depending only on Δ such that if p is a large prime and p ≡ 1(mod D^{∗}(Δ)), then the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides generically with its lower bound, i.e., for such p, H_{p}(Δ) is Zariski dense in M_{p}(Δ) .

This theorem shows that the Adolphson-Sperber conjecture is not too far from being true. Their conjecture is equivalent to the statement that one can take D^{∗}(Δ) to be D(Δ) (the ideal case). Our main results are several decomposition theorems. These decomposition theorems imply Theorem 3. They can also be used to prove that the Adolphson-Sperber conjecture is true in various special cases. In this direction, we have the following results.

Theorem 4. Conjecture 1 is true for n = 2.

For applications to zeta functions, we have

Theorem 5. For a general projective hypersurface X of degree d, the Newton polygon of the interesting part of the zeta function Z(X, t) coincides with its lower bound (this gives a p-adic proof of the Dwork-Mazur conjecture).

Theorem 6. For a general affine hypersurface X of degree d, the Newton polygon of the interesting part (if it is a polynomial) of the zeta function Z(X, t) coincides with its lower bound.

Theorem 7. Let X be a general affine hypersurface of degree d_{i} with respect to x_{i} (1 ≤ i ≤ n). Then, the Newton polygon of the interesting part (if it is a polynomial) of the zeta function Z(X, t) coincides with its lower bound.

A harder question is to classify all large prime numbers p for which H_{p}(Δ) is Zariski dense in M_{p}(Δ). In the last section, we give a conjectural answer to this question which is related to Frobenius coverings of lattice points.

We now sketch the main ideas in proving Theorem 3. In the above, it is mentioned that H_{p}(Δ) is a Zariski open subset of M_{p}(Δ) (possibly empty). Thus, Theorem 3 is reduced to finding a single example (for each Δ) with desired Newton polygon. If Δ is a simplex, then the “diagonal forms” provide the desired examples. Thus, Theorem 3 is true for a simplex. For a general Δ, a natural question is whether or not there are decomposition theorems which successively reduce a general Δ to a simpler Δ, and eventually to a simplex. Depending on how strong we want the decomposition theorems to be, we have different answers. Ideally, one would hope that L^{∗}(f, t) decomposes as a product of the L-functions corresponding to the restrictions of f to various pieces in a decomposition of Δ. We shall give such a theorem corresponding to the trivial boundary decomposition of Δ. This theorem does not reduce Theorem 3 from a general Δ to a simpler Δ, but it is useful sometimes (for example, in reducing a projective case to an affine case).

If we relax our requirments on the decomposition theorems a little and are concerned only that the Newton polygon coincides with its lower bound, then there is a non-trivial decomposition theorem obtained in [15]. This theorem reduces Theorem 3 from a general Δ to a simpler Δ having only one codimension 1 face which does not contain the origin. To describe this more precisely, let σ be an (n − 1)-dimensional closed face of Δ which does not contain the origin and let f_{σ} be the restriction of f to the closed polyhedron generated by σ and the origin. If f is nondegenerate, then f_{σ} is nondegenerate. Thus, the theorem of Adolphson and Sperber shows that L^{∗}(f_{σ}, t)^{(−1)^{n−1}} is a polynomial, whose Newton polygon lies above its lower bound P (σ). The main result of [15] is the following facial decomposition theorem.

Theorem 8. Let f be non-degenerate. Then the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides with its lower bound P (Δ) if and only if the Newton polygon of L^{∗}(f_{σ}, t)^{(−1)^{n−1}} coincides with its lower bound P (σ) for all (n − 1)-dimensional closed faces σ which do not contain the origin.

If we further relax our requirements on the decomposition theorems and are concerned only that the Newton polygon coincides generically with its lower bound, then there are further nontrivial decomposition theorems. In this paper, we establish two such theorems corresponding to two types of decompositions of Δ. The first type is the star decomposition of Δ with respect to a lattice point in Δ. In this context, Theorem 8 can be viewed as a special case, corresponding to the star decomposition with respect to the origin. The second type is the hyperplane decomposition obtained by a hyperplane cut of Δ. The main technical idea is to introduce a degree polygon which involves the degree of the leading form of the Hasse polynomial h_{p}(Δ) in terms of certain variables. This degree polygon is finer than the Newton polygon. We shall give an upper bound for the degree polygon and prove decomposition theorems about the coincidence of the degree polygon with its upper bound for the two types of decompositions of Δ. These decompositions are fine enough to reduce a general Δ to a simplicial Δ. This immediately implies Theorem 3. The proof of Theorems 5-7 is obtained by a suitable sequence of combinations of the star decomposition and the hyperplane decomposition so that D^{∗}(Δ) = 1. In particular, the proof of Theorem 5 uses a decomposition construction on resolution of singularities of toric varieties. One might expect a deeper and more general connection bewteen generic Newton polygons and resolution of singularities.

As in [15], we shall work on the chain level and avoid any direct use of Dwork cohomology. Only at the last step will we appeal to the theorem of Adolphson and Sperber about Dwork cohomology which asserts that L^{∗}(f, t)^{(−1)^{n−1}} is a polynomial of degree n!V(Δ). This will make the paper more elementary and self-contained. Another advantage of working on the chain level is that we do not need any non-degeneracy assumptions. Thus, the results hold for all f and all prime numbers p. All of the above results hold for a more general L-function L(f, t) studied by Adolphson and Sperber [2], see §8. Adolphson and Sperber [3] have recently extended their conjecture to twisted exponential sums. Our counterexample shows that this more general conjecture is also false. We expect that many of the results in the present paper generalize to these twisted sums.

Acknowledgment. I would like to thank Neal Koblitz for his detailed comments, Branko Grünbaum for consultations on convex polytopes and Jeffrey Lagarias for suggesting the reference [8]. I would also like to thank Alan Adolphson and the referee for several key comments and corrections. Part of the materials here was presented in the number theory seminar at the University of California at Berkeley in 1992. I would like to thank Hendrik Lenstra for his support and hospitality.

1. Counterexamples

In this section, we prove Theorem 2 by giving counterexamples of Conjecture 1. We begin with some observations on diagonal forms. Let 0, V_{1}, ⋯, V_{n} be the vertices of an n-dimensional integral simplex Δ. Consider the solutions of the equation

∑_{j=1}^{n} u_{j}V_{j} ≡ 0 (mod 1), 0 ≤ u_{j} < 1, u_{j} rational.  (1.1)

There is a Frobenius action on the solutions of (1.1). If u = (u_{1}, ⋯, u_{n}) is a solution of (1.1), then the Frobenius action {pu} = ({pu_{1}}, ⋯, {pu_{n}}) is also a solution of (1.1), where {u_{i}} is defined to be the reduction of u_{i} modulo 1. In other words, {u_{i}} is simply the fractional part of u_{i}. Define the weight of u to be ∑_{i=1}^{n} u_{i}. We say that the weight function is stable under the Frobenius action if for all solutions u of (1.1),

∑_{i=1}^{n} u_{i} = ∑_{i=1}^{n}{pu_{i}}.

Let f(x) be the generic diagonal form

f(x) = ∑_{j=1}^{n} a_{j}x^{V_{j}}.

The explicit computation in [15] and the Stickelberger relation show that the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides with its lower bound if and only if the weight function is stable under the Frobenius action. We now use this fact to construct counterexamples of Conjecture 1.

Let us consider the 6-dimensional simplex Δ whose non-zero vertices are the column vectors of the following matrix:

M = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{matrix} )

Denote these non-zero vertices by V_{1}, ⋯, V_{6}. Clearly, all of the non-zero vertices lie on the hyperplane x_{1} = 1. It follows that D(Δ) = 1. One checks that there are no lattice points on Δ except for the vertices. Thus, the generic Laurent polynomial for this simplex is simply the generic diagonal form

f(x) = ∑_{j=1}^{6} a_{j}x^{V_{j}},

where we have omitted the constant term in f(x) since it does not affect the Newton polygon. The determinant of the matrix M is easily seen to be 3. Thus, L^{∗}(f, t)^{−1} is a polynomial of degree 3, divisible by (1 − t). There are only three lattice points contained in the fundamental domain R^{6}/ZV_{1} + ⋯ + ZV_{6}. One of them is the origin. The other two are

(2, 1, 1, 1, 1, 1) = ∑_{i=1}^{6} \frac{1}{3}V_{i}, (4, 2, 2, 2, 2, 2) = ∑_{i=1}^{6} \frac{2}{3}V_{i}.  (1.2)

These two lattice points have weight 2 and 4, respectively. The Adolphson-Sperber lower bound is the polygon in the plane with vertices (0, 0), (1, 0), (2, 2) and (3, 6), whose first non-zero slope is 2. If p ≡ 2 (mod 3), then one checks that the Frobenius map takes any one of the points in (1.2) to the other point. Thus, the weight function is not stable under the Frobenius action. This shows that the generic Newton polygon lies strictly above its lower bound. In fact, in this case the generic Newton polygon of L^{∗}(f, t)^{−1} is the polygon in the plane with vertices (0, 0), (1, 0) and (3, 6), whose first non-zero slope is 3. This proves that Conjecture 1 is false if n = 6.

To construct counterexamples in higher dimensions, we can simply let Δ be the integral simplex generated by the origin and the column vectors of the matrix

M^{∗} = ( \begin{matrix}M & B \\ 0 & I \end{matrix} ) ,

where M is the above matrix of order 6, I is the identity matrix of order (n − 6), the first row of B is (1, 1, ⋯, 1) and all other rows of B are zero. Thus, the first row of M^{∗} is (1, 1, ⋯, 1) and D(Δ) = 1. One checks that there are no lattice points on Δ other than the vertices. However, if p ≡ 2 (mod 3), the above argument shows that the generic Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} lies strictly above its lower bound.

A less obvious counterexample in dimension 5 is the simplex whose non-zero vertices are the column vectors V_{i} of the following matrix:

M′ = ( \begin{matrix}1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \end{matrix} )

Clearly, D(Δ) = 1. One computes that the determinant of the matrix M′ has absolute value 3. A direct proof shows that there are no lattice points on Δ except for the vertices. Thus, L^{∗}(f, t) is a polynomial of degree 3, divisible by (1 − t). There are only three lattice points contained in the fundamental domain R^{5}/ZV_{1} + ⋯ + ZV_{5}. One of them is the origin. The other two are

(2, 1, 1, 1, 1) = \frac{2}{3}V_{1} + ∑_{i=2}^{5} \frac{1}{3}V_{i}, (3, 2, 2, 2, 2) = \frac{1}{3}V_{1} + ∑_{i=2}^{5} \frac{2}{3}V_{i}.  (1.3)

These two lattice points have weight 2 and 3, respectively. One checks that if p ≡ 2 (mod 3), then the weight function is not stable under Frobenius action. Thus, the generic Newton polygon lies strictly above its lower bound. Theorem 2 is proved.

Let D^{∗} be the smallest positive integer for which Conjecture 1 is true. We can construct examples which show that the quotient D^{∗}/D(Δ) is unbounded as n varies. Thus, D^{∗} depends not only on D(Δ) but also on the geometric shape of Δ.

§2. Dwork’s p-Adic Theory

In this section, following Adolphson and Sperber [1] [2] (with some paraphrasing), we shall present their version of Dwork’s p-adic theory describing L-functions in terms of the Fredholm determinant of a certain nuclear infinite matrix. We shall prove a new theorem (Theorem 2.4) concerning conditions under which the Newton polygon coincides with its lower bound. This result is the basis of our later discussions.

Let Q_{p} be the field of p-adic numbers, and let Ω be the completion of an algebraic closure of Q_{p}. Let K denote the unramified extension of Q_{p} in Ω of degree a, where q = p^{a}. Let π ∈ Ω satisfy π^{p−1} = −p. Then Ω_{1} = Q_{p}(π) is a totally ramified extension of Q_{p} of degree p − 1, in fact, Ω_{1} = Q_{p}(ζ_{p}), where ζ_{p} is a primitive p-th root of unity. Let Ω_{a} be the compositum of Ω_{1} and K. Then Ω_{a} is an unramified extension of Ω_{1} of degree a. The residue fields of Ω_{a} and K are both F_{q}, and the residue fields of Ω_{1} and Q_{p} are both F_{p}. The Frobenius automorphism x ↦ x^{p} of Gal(F_{q}/F_{p}) lifts to a generator τ of Gal(K/Q_{p}) which is extended to Ω_{a} by requiring that τ(π) = π. If ζ is a (q − 1)-st root of unity in Ω_{a}, then τ(ζ) = ζ^{p}. Denote by “ord” the additive valuation on Ω normalized by ord p=1, and denote by “ord_{q}” the additive valuation on Ω normalized by ord_{q}q=1.

Let E(t) be the Artin-Hasse exponential series:

E(t) = exp( ∑_{m=0}^{∞} \frac{t^{p^{m}}}{p^{m}}) ∈ (Z_{p} ∩ Q)[[t]].  (2.1)

Let γ ∈ Ω_{1} be a root of ∑_{m=0}^{∞} \frac{t^{p^{m}}}{p^{m}} = 0 satisfying ordγ = ordπ = \frac{1}{p−1}. The series

θ(t) = E(γt) = ∑_{m=0}^{∞} λ_{m}t^{m}  (2.2)

is a splitting function in Dwork’s terminology, and its coefficients satisfy

ordλ_{m} ≥ \frac{m}{p − 1}, λ_{m} ∈ Ω_{1},  (2.3)
ordλ_{m} = \frac{m}{p − 1} for 0 ≤ m ≤ p − 1.  (2.4)

Write V = (v_{1}, ⋯, v_{n}) ∈ Z^{n}, x = (x_{1}, ⋯, x_{n}), x^{V} = x_{1}^{v_{1}} ⋯ x_{n}^{v_{n}}. For a Laurent polynomial f(x_{1}, ⋯, x_{n}) ∈ F_{q}[x_{1}, x_{1}^{−1}, ⋯, x_{n}, x_{n}^{−1}], write

f = ∑_{j=1}^{J} ā_{j}x^{V_{j}},  (2.5)

where V_{j} ∈ Z^{n} and the ā_{j}’s are non-zero elements of F_{q}. Let a_{j} be the Teichmüller lifting of ā_{j} in Ω, i.e., a_{j}^{q} = a_{j}. Set

F (f, x) = ∏_{j=1}^{J} θ(a_{j}x^{V_{j}})  (2.6)
F_{a}(f, x) = ∏_{i=0}^{a−1} F^{τ^{i}}(f, x^{p^{i}}).  (2.7)

Note that (2.3) implies that F (f, x) and F_{a}(f, x) are well defined as formal Laurent series in x_{1}, ⋯, x_{n} with cofficients in Ω_{a}.

To describe the growth conditions satisfied by F, write

F (f, x) = ∑_{r∈Z^{n}} F_{r}(f)x^{r}.  (2.8)

Then from (2.2) and (2.6), one checks that

F_{r}(f) = ∑_{u}(∏_{j∈J} λ_{u_{j}}a_{j}^{u_{j}}),  (2.9)

where the outer sum is over all solutions of the linear system

∑_{j=1}^{J} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} integral.  (2.10)

Thus, F_{r}(f) = 0 if (2.10) has no solutions. Otherwise, (2.3) implies that

ord(F_{r}(f)) ≥ \frac{1}{p − 1} inf_{u}\{∑_{j=1}^{J} u_{j}\},  (2.11)

where the inf is taken over all solutions of (2.10).

For a rational point r ∈ Z^{n}, Adolphson and Sperber [1] defined a weight function w(r) by

w(r) = inf_{u}{∑_{j=1}^{J} u_{j}},  (2.12)

where now the inf is taken over all nonnegative rational solutions of (2.10). Then (2.11) implies that

ord(F_{r}(f)) ≥ \frac{w(r)}{p − 1},  (2.13)

with the obvious convention that F_{r}(f) = 0 if w(r) = +∞.

The weight function can be described geometrically. Let Δ(f) be the convex hull in R^{n} generated by the V_{j} and the origin. This is called the Newton polyhedron of f. Let C(Δ) denote the cone generated by Δ. Then C(Δ) is the union of all rays emanating from the origin and passing through Δ(f). Equation (2.10) has a solution u whose components are nonnegative rational numbers if and only if r ∈ C(Δ). Thus, w(r) = +∞ if and only if r ∉ C(Δ). If r ∈ C(Δ), the ray emanating from the origin and passing through r intersects Δ(f) in a face that does not contain the origin. Let ∑_{i=1}^{n} α_{i}X_{i} = 1 be the equation of a hyperplane passing through this face (this hyperplane is not uniquely determined unless the face has dimension n − 1). Then, by standard arguments in linear programming,

w(r) = ∑_{i=1}^{n} α_{i}r_{i}.  (2.14)

Let D(Δ) be the least common denominator of all the α_{i}’s over all the n − 1-dimensional faces of Δ(f) which do not contain the origin. Then

w(Z^{n}) ⊆ \frac{1}{D(Δ)}Z_{≥0} ∪ \{+∞\},  (2.15)

where Z_{≥0} denotes the set of nonnegative integers. From the above description of w(r), it is immediately seen that

Lemma 2.1. (a) For any nonnegative rational number k, w(kr) = kw(r).

(b) w(r + r′) ≤ w(r) + w(r′), with equality holding if and only if r and r′ are cofacial, i.e., w(r)^{−1}r and w(r′)^{−1}r′ lie on the same closed face (of dimension at most n − 1) of Δ.

(c) Let u = (u_{1}, ⋯, u_{J}) be a rational solution of (2.10). Let u_{j_{i}} be its non-zero coordinates. Then w(r) = ∑_{j} u_{j} if and only if the non-zero lattice points among the V_{j_{i}} lie on the same closed face of codimension 1 not containing the origin.

Let M(f) be the set of lattice points in the cone C(Δ). For real numbers b and c with 0 < b ≤ p/(p − 1), define the following spaces of p-adic functions:

\begin{aligned}L(b, c) & = \{ ∑_{r∈M(f)} C_{r}x^{r} | C_{r} ∈ Ω_{a}, ord_{p}C_{r} ≥ bw(r) + c\} \\ L(b) & = ∪_{c∈R}L(b, c). \end{aligned}

One checks that

F (f, x) ∈ L(\frac{1}{p − 1}, 0), F_{a}(f, x) ∈ L(\frac{p}{q(p − 1)}, 0).

Define an operator ψ on formal Laurent series by

ψ( ∑_{r∈M(f)} C_{r}x^{r}) = ∑_{r∈M(f)} C_{pr}x^{r}.

It is immediate from Lemma 2.1(a) that ψ(L(b, c)) ⊂ L(pb, c). It follows that the operator α_{a} = ψ^{a} ∘F_{a}(f, x) is an Ω_{a}-linear endomorphsim of the space L(b) and the operator α_{1} = τ^{−1} ∘ψ∘F (f, x) is an Ω_{a}-semilinear (τ^{−1}-linear) endomorphsim of the space L(b). The operators α_{a}^{m} and α_{1}^{m} have well defined traces. The Dwork trace formula asserts that

S_{m}^{∗}(f) = (q^{m} − 1)^{n}Tr(α_{a}^{m}).

One checks that

α_{1}^{a} = α_{1}^{a−2}τ^{−1} ∘ ψ ∘ F (f, x) ∘ τ^{−1} ∘ ψ ∘ F (f, x)
= α_{1}^{a−2}τ^{−1} ∘ ψ ∘ τ^{−1} ∘ F^{τ} (f, x) ∘ ψ ∘ F (f, x)
= α_{1}^{a−2}(τ^{−1})^{2} ∘ ψ^{2} ∘ F^{τ} (f, x^{p}) ∘ F (f, x)
= ⋯ = ψ^{a} ∘ F_{a}(f, x) = α_{a}.

We now describe the matrix form of the operators α_{1} and α_{a}. Fix a choice π^{1/D} of D-th root of π in Ω. Define a space of functions

B = { ∑_{r∈M(f)} C_{r}π^{w(r)}x^{r} | C_{r} ∈ Ω_{a}(π^{1/D}), C_{r} → 0 as r → ∞}.

Then, the monomials π^{w(r)}x^{r} form an orthonormal basis of the p-adic Banach space B. Furthermore, if b > 1/(p − 1), then L(b) ⊆ B. The operator α_{a} (resp. α_{1}) is an Ω_{a}-linear (resp. Ω_{a}- semilinear ) nuclear endomorphsim of the space B. Let Γ be the orthonormal basis {π^{w(r)}x^{r}}_{r∈M(f)} written as a column vector. The operator α_{1} is given by α_{1}Γ = A_{1}(f)Γ, where A_{1}(f) is the infinite matrix whose rows are indexed by s and columns are indexed by r.

A_{1}(f) = (a_{s,r}(f)) = (F_{ps−r}(f)π^{(w(r)−w(s))}).  (2.16)

Since α_{a} = α_{1}^{a} and α_{1} is τ^{−1}-linear, the operator α_{a} is given by

α_{a}Γ = α_{1}^{a}Γ = α_{1}^{a−1}A_{1}Γ = α_{1}^{a−2}A_{1}^{τ^{−1}}A_{1}Γ = A_{1}^{(τ^{−1})^{a−1}} ⋯ A_{1}^{τ^{−1}}A_{1}Γ.

To simplify notation, let

A_{a}(f) = ∏_{i=0}^{a−1} A_{1}(f)^{τ^{i}} = A_{1}A_{1}^{τ^{1}} ⋯ A_{1}^{τ^{a−1}}.  (2.17)

Then, the matrix of α_{a} under the basis Γ is A_{a}^{τ^{−1}}(f). Since we will work exclusively with matrices later on, for simplicity we call A_{1}(f) = (a_{s,r}(f)) the infinite semilinear Frobenius matrix and A_{a}(f) the infinite linear Frobenius matrix.

By (2.13), (2.16) and Lemma 2.1, we obtain the estimate

orda_{s,r}(f) ≥ \frac{w(ps − r) + w(r) − w(s)}{p − 1} ≥ w(s).  (2.18)

Thus, A_{1}(f) and A_{a}(f) are nuclear matrices. They all have well defined traces. In addition, the Fredholm determinants det(I − tA_{a}) and det(I − tA_{1}) are well defined and p-adically entire (i.e., convergent for all t ∈ Ω). The Dwork trace formula asserts that

S_{m}^{∗}(f) = (q^{m} − 1)^{n}Tr^{τ^{−1}}(A_{a}(f)^{m}).  (2.19)

This form is slightly different from the usual form because we want to work with A_{a}(f) instead of the Frobenius matrix A_{a}^{τ^{−1}}(f) in order to simplify notations later on. The nontrivial additive character Ψ implicit in the left side of (2.19) (see (0.1)) is given by

Ψ(t) = θ(1)^{t}  (2.20)

for t ∈ F_{p} (θ(1) is a primitive p-th root of unity). An equivalent form of (2.19) is

L^{∗}(f, t)^{(−1)^{n−1}} = ∏_{i=0}^{n} det^{τ^{−1}}(I − tq^{i}A_{a}(f))^{(−1)^{i}\binom{n}{i}},  (2.21)

where

det(I − tA_{a}(f)) = exp(− ∑_{m=1}^{∞} \frac{Tr(A_{a}(f)^{m})t^{m}}{m} ).  (2.22)

If L^{∗}(f, t)^{(−1)^{n−1}} is a polynomial, then (2.21) shows that the determination of the Newton polygon of det(I − tA_{a}(f)) will enable us to determine the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}}. Define a weight function W (k) on non-negative integers as follows:

W (k) = card\{r ∈ Z^{n} | w(r) = \frac{k}{D(Δ)}\}.  (2.23)

Let P (Δ) be the Newton polygon with vertices (0, 0) and

(∑_{k=0}^{m} W (k), \frac{1}{D(Δ)} ∑_{k=0}^{m} kW (k)), m = 0, 1, 2, · · · .  (2.24)

The lower bound of Adolphson and Sperber [1] is

Proposition 2.2. The Newton polygon of det(I −tA_{a}(f)) computed with respect to q lies above P (Δ).

Let ξ ∈ Ω be such that ξ^{D} = π^{p−1} = −p; then ordξ = 1/D. By (2.18), A_{1}(f) has the block form

A_{1}(f) = ( \begin{matrix}A_{00} & A_{01} & … & A_{0i} & … \\ ξ^{1}A_{10} & ξ^{1}A_{11} & … & ξ^{1}A_{1i} & … \\ ⋮ & ⋮ & ... & ⋮ \\ ξ^{i}A_{i0} & ξ^{i}A_{i1} & … & ξ^{i}A_{ii} & … \\ ⋮ & ⋮ & ... & ⋮ \end{matrix} ) ,  (2.25)

where the A_{ij}’s are matrices of W (i) rows and W (j) columns whose entries are p-adic integers in Ω. This block form implies

Proposition 2.3. The Newton polygon of det(I −tA_{1}(f)) computed with respect to p lies above P (Δ).

In general, we cannot expect that the Newton polygons of det(I − tA_{a}(f)) and det(I − tA_{1}(f)) are the same (even though they have the same lower bound). For a counterexample in the case of F-crystals, see Katz [7]. However, if we are only interested in the question of whether or not the Newton polygon of det(I − tA_{a}(f)) coincides with its lower bound, the following theorem shows that we can restrict ourselves to det(I − tA_{1}(f)). This result is used in [15] without proof.

Theorem 2.4. The Newton polygon of det(I − tA_{a}(f)) computed with respect to q coincides with P (Δ) if and only if the Newton polygon of det(I −tA_{1}(f)) computed with respect to p coincides with P (Δ).

Proof. We prove the theorem by induction on the sides. The first side of the Newton polygon of det(I − tA_{1}(f)) coincides with the first side of P (Δ) if and only if det A_{00} ≢ 0 (mod ξ). To simplify notations, we use the convention that mod ξ (or mod p) means the reduction modulo the maximal ideal in the ring of integers in Ω. We can write

A_{a}(f) ≡ ( \begin{matrix}A_{00} ⋯ A_{00}^{τ^{a−1}} & ∗ \\ 0 & 0 \end{matrix} )(modξ).  (2.26)

It follows that the first side of the Newton polygon of det(I − tA_{a}(f)) coincides with the first side of P (Δ) if and only if det A_{00} ≢ 0 (mod ξ). Thus, the theorem is true for the first side.

Assume that det A_{00} ≢ 0 (mod ξ). We now use a triangulation procedure to prove the theorem for the first two sides. Form an elementary nuclear matrix B (to be determined) as follows:

B = ( \begin{matrix}I_{00} & 0 & … & 0 & … \\ ξ^{1}B_{10} & I_{11} & … & 0 & … \\ ⋮ & ⋮ & ... & ⋮ \\ ξ^{i}B_{i0} & 0 & … & I_{ii} & … \\ ⋮ & ⋮ & ... & ⋮ \end{matrix} ) .  (2.27)

Its inverse is

B^{−1} = ( \begin{matrix}I_{00} & 0 & … & 0 & … \\ −ξ^{1}B_{10} & I_{11} & … & 0 & … \\ ⋮ & ⋮ & ... & ⋮ \\ −ξ^{i}B_{i0} & 0 & … & I_{ii} & … \\ ⋮ & ⋮ & ... & ⋮ \end{matrix} ) ,  (2.28)

where B_{i0} is a matrix of W (i) rows and W (0) columns, and I_{ii} is the identity matrix of order W (i). Now the matrix BA_{1}B^{−τ} has the form

BA_{1}B^{−τ} = ( \begin{matrix}C_{00} & C_{01} & … & C_{0i} & … \\ ξ^{1}C_{10} & ξ^{1}C_{11} & … & ξ^{1}C_{1i} & … \\ ⋮ & ⋮ & ... & ⋮ \\ ξ^{i}C_{i0} & ξ^{i}C_{i1} & … & ξ^{i}C_{ii} & … \\ ⋮ & ⋮ & ... & ⋮ \end{matrix} ) ,  (2.29)

where

C_{00} = A_{00} − ξ^{1}A_{01}B_{10}^{τ} − ⋯ − ξ^{k}A_{0k}B_{k0}^{τ} − ⋯,
C_{i0} = B_{i0}A_{00} + A_{i0} − ξ^{1}{B_{i0}A_{01} + A_{i1}}B_{10}^{τ} − ⋯ − ξ^{k}{B_{i0}A_{0k} + A_{ik}}B_{k0}^{τ} − ⋯ (i > 0),
C_{0j} = A_{0j}, C_{ij} = B_{i0}A_{00} + A_{ij} (i ≥ 1, j ≥ 1).  (2.30)

We want to choose suitable matrices B_{i0} such that

C_{i0} = 0 for all i ≥ 1.  (2.31)

Since A_{00} is invertible, (2.31) can be rewritten as

B_{i0} = −A_{i0}A_{00}^{−1} + ξ^{1}{B_{i0}A_{01} + A_{i1}}B_{10}^{τ}A_{00}^{−1} + ⋯
+ξ^{k}{B_{i0}A_{0k} + A_{ik}}B_{k0}^{τ}A_{00}^{−1} + · · · .  (2.32)

Thus, by successive iteration of (2.32) or by Hensel’s lifting lemma or by the fixed point theorem for a contraction map in a Banach space, we deduce that there are solution matrices B_{i0} whose elements are p-adic integers invariant under the action of τ^{a}.

If we go through the above triangulation procedure again but ignore the action of τ in (2.30) and (2.32), we conclude that there is a nuclear matrix B_{1} of the form (2.27), whose entries are p-adic integers, such that

B_{1}A_{1}(f)B_{1}^{−1} = ( \begin{matrix}D_{00} & D_{01} & … & D_{0i} & … \\ 0 & ξ^{1}D_{11} & … & ξ^{1}D_{1i} & … \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & ξ^{i}D_{i1} & … & ξ^{i}D_{ii} & … \\ ⋮ & ⋮ & ... & ⋮ \end{matrix} ) ,  (2.33)

where D_{ij} is a matrix of the same size as C_{ij}. It follows from (2.30) that C_{ij} − D_{ij} is divisible by ξ. Comparing (2.33) with (2.29), we conclude that the Newton polygon of det(I − tA_{1}(f)) coincides with P (Δ) if and only if the Newton polygon of det(I − tBA_{1}(f)B^{−τ}) coincides with P (Δ). Since

BA_{a}(f)B^{−1} = BA_{a}(f)B^{−τ^{a}} = (BA_{1}(f)B^{−τ}) ⋯ (BA_{1}(f)B^{−τ})^{τ^{a−1}},

using the triangular form obtained above, we deduce that the first two sides of the Newton polygon of det(I −tA_{1}(f)) computed with respect to p coincide with P (Δ) if and only if the first two sides of the Newton polygon of det(I − tA_{a}(f)) computed with respect to q coincide with P (Δ). Repeating the above argument, by induction we see that the theorem is true. Actually, the rationality of L^{∗}(f, t) and the trivial archimedean estimate on its zeros and poles imply that we only need to prove the theorem for the first nD sides.

We shall use the convention that the first m sides of P (Δ) mean the sides in P (Δ) with slopes at most m/D. Similarly, the m^{th} side in P (Δ) means the side with slope equal to m/D (even though this side may not exist). The above proof implies

Corollary 2.5. The Newton polygon of det(I − tA_{a}(f)) computed with respect to q coincides with P (Δ) for the first m sides if and only if the Newton polygon of det(I − tA_{1}(f)) computed with respect to p coincides with P (Δ) for the first m sides.

Let H_{p}(Δ, m) (resp. H_{p}(Δ, ≤ m)) be the moduli space of f ∈ M_{p}(Δ) for which the m^{th} side (resp. the first m sides) of the Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound. Let A(m) be the matrix of the first ∑_{k≤m} W (k/D) rows and columns in A_{1}(f). The space H_{p}(Δ, m) is simply the compliment of the hypersurface defined by the determinant equation

h_{p}(Δ, m) = \frac{det(A(m))}{ξ^{∑_{k≤m} \frac{k}{D} W ( \frac{k}{D})}} ≡ 0 (mod ξ).

Thus, a defining equation for H_{p}(Δ, m) can be computed in a finite amount of time. We shall call h_{p}(Δ, m) the Hasse polynomial of the first m^{th} side. Similarly, the moduli space H_{p}(Δ, ≤ m) is simply the intersection of the moduli spaces H_{p}(Δ, k) (0 ≤ k ≤ m). Its compliment is defined by the product polynomial h_{p}(Δ, ≤ m) = ∏_{k≤m} h_{p}(Δ, k). Very little is known about these polynomials or the Hasse variety they define. Whether these polynomials are non-zero or not is the content of Conjecture 1. In order to prove that these polynomials are non-vanishing in certain cases, we shall study the stronger question of whether or not the leading form in certain variables vanishes. This is accomplished by introducing certain maximizing functions from linear programming and investigating certain degree polygons defined by these maximizing functions.

§3. Maximizing Functions on Rational Points

In this section, we assume that the origin 0 is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin (for a more general Δ, Theorem 8 in §0 reduces it to the present case).

Let V_{1}, ⋯, V_{J} be J fixed lattice points on the face Δ_{1} such that the V_{j} include all the vertices of Δ_{1}. Then Δ is generated by 0 and the V_{j}. Let C(Δ) be the cone generated by Δ.

Definition 3.1. For a rational point r ∈ C(Δ), define

m(V_{1}, Δ; r) = sup{u_{1} | ∑_{j=1}^{J} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} rational}.

This function is defined with respect to V_{1} (not necessarily a vertex). One can similarly define m(V_{j}, Δ; r) with respect to any other lattice point V_{j}. In the following, V_{1}r denotes the line passing through the points V_{1} and r. If r lies on the line V_{1}0, then V_{1}r = V_{1}0. The object of this section is to determine when a solution of the equation

∑_{j=1}^{J} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} rational,  (3.1)

makes u_{1} maximal. Let Δ^{∗} = Δ^{∗}(V_{1}, r) be the face of largest codimension of Δ_{1} such that Δ^{∗} contains the point of intersection of the ray joining V_{1} and \frac{1}{w(r)}r and the boundary of Cone(Δ). If r lies on the line V_{1}0, we simply define Δ^{∗}(V_{1}, r) to be the lattice point V_{1}.

Lemma 3.2. (i). Let (u_{1}, ⋯, u_{J}) be a solution of (3.1). Let u_{j_{1}}, ⋯, u_{j_{k}} (all j_{i} ≥ 2) be its non-zero coordinates (thus, u_{1} is not included). Then m(V_{1}, Δ; r) = u_{1} if and only if the lattice points V_{j_{1}}, ⋯, V_{j_{k}} belong to Δ^{∗}(V_{1}, r), (i.e., the lattice points V_{j_{1}}, ⋯, V_{j_{k}} are cofacial with respect to V_{1}, which means that they lie on the same closed face of Δ not containing V_{1}).

(ii). Let r be a rational point in the cone C(Δ). Let Δ′(r) be the polyhedron generated by V_{1}, the origin and Δ^{∗}(V_{1}, r). Then, the following identity is true:

m(V_{1}, Δ; r) = m(V_{1}, Δ′(r); r).

Proof. We may assume that r ≠ 0. Since the V_{j} lie on the unique face Δ_{1} of codimension 1 not containing the origin, Lemma 2.1 shows that the sum ∑_{j} u_{j} = w(r) depends only on r. Dividing the u_{j} by w(r), we may assume that w(r) = 1. Then r lies on Δ_{1}. Under these assumptions, equation (3.1) can be rewritten as

r − V_{1} = ∑_{j=2}^{J} u_{j}(V_{j} − V_{1}).

Since u_{1} + ⋯ + u_{J} = 1, maximizing u_{1} is equivalent to minimizing the sum ∑_{j=2}^{J} u_{j}. The latter is simply the weight function of Adolphson and Sperber defined with respect to the (n−1)-dimensional space spanned by the V_{j} − V_{1} (all j ≥ 2). Part (c) of Lemma 2.1 implies that the sum ∑_{j=2}^{J} u_{j} is minimal if and only if the V_{j_{i}} − V_{1} are cofacial with respect to the origin. This is equivalent to saying that the V_{j_{i}} are cofacial with respect to V_{1}. This proves (i). It is easy to see that (ii) is a restatement of (i). The lemma is proved.

Lemma 3.3. (i). Let r be a rational point in the cone C(Δ). Then, m(V_{1}, Δ; pr) ≥ pm(V_{1}, Δ; r) for all positive rational numbers p. (ii). Let r_{1} and r_{2} be two rational points in the cone C(Δ). Then m(V_{1}, Δ; r_{1} + r_{2}) ≥ m(V_{1}, Δ; r_{1}) + m(V_{1}, Δ; r_{2}), with equality holding if and only if r_{1} and r_{2} are cofacial with respect to V_{1}, i.e., the rays emanating from V_{1} and passing through r_{1} and r_{2} hit the same closed face of C(Δ) not containing V_{1}.

Proof. One can easily check (i). We now prove (ii). Let

r_{1} = u_{1}V_{1} + ⋯ + u_{J}V_{J}, u_{1} = m(V_{1}, Δ; r_{1}),

r_{2} = w_{1}V_{1} + ⋯ + w_{J}V_{J}, w_{1} = m(V_{1}, Δ; r_{2}).

Then,

r_{1} + r_{2} = (u_{1} + w_{1})V_{1} + ⋯ + (u_{J} + w_{J})V_{J}.  (3.2)
m(V_{1}, Δ; r_{1}) + m(V_{1}, Δ; r_{2}) = u_{1} + w_{1} ≤ m(V_{1}, Δ; r_{1} + r_{2}).  (3.3)

Let V_{j_{i}} (all j_{i} ≥ 2) be the lattice points with non-zero coefficients in (3.2). By Lemma 3.2, the inequality in (3.3) is an equality if and only if the V_{j_{i}} are cofacial with respect to V_{1}. This happens if and only if r_{1} and r_{2} are cofacial with respect to V_{1}. The lemma is proved.

More generally, we can define a maximizing function on rational points with respect to several lattice points.

Definition 3.4. Let T be a subset of the V_{j}. For a rational point r ∈ C(Δ), define

m(T, Δ; r) = sup{ ∑_{V_{j}∈T} u_{j} | ∑_{j=1}^{J} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} rational}.

This is a standard maximizing function in linear programming. For given r, there are various algorithms to compute m(T, Δ; r). The simplex method in linear programming shows that the set of optimal solutions is a “face” formed by some of the lattice points V_{j}. For our purpose, we need to describe the optimal face in various special cases. A trivial case is when T contains all the V_{j}. In this case, m(T, Δ; r) = w(r) (the weight function of Adolphson-Sperber). In the above, we treated the case T = V_{1}. For latter applications, we now study another case in which T is obtained by a hyperplane cut of Δ.

Let H be a hyperplane passing through the origin such that the intersection of H with Δ is an integral polyhedron of dimension n − 1 whose vertices are contained in the V_{j}. Then H cuts Δ into two pieces Δ(1) and Δ(2). We may suppose that both Δ(1) and Δ(2) are closed. Then Δ is the union of Δ(1) and Δ(2). Let T be the set of the V_{j} which are contained in Δ(1), and let T_{H} be the intersection of T and H. The following lemma tells when a solution of (3.1) attains the sup in Definition 3.4.

Lemma 3.5. Let (u_{1}, ⋯, u_{J}) be a solution of (3.1). Suppose that u_{j_{1}}, ⋯, u_{j_{k}} are its non-zero coordinates. (i). If r ∈ C(Δ(1)), then m(T, Δ; r) = ∑_{V_{j}∈T} u_{j} if and only if Δ(1) contains all the lattice points V_{j_{i}}. (ii). If r ∈ C(Δ(2)) and m(T, Δ; r) = ∑_{V_{j}∈T} u_{j}, then Δ(2) contains all the lattice points V_{j_{i}}. (iii). If r ∈ H and m(T, Δ; r) = ∑_{V_{j}∈T} u_{j}, then H contains all the lattice points V_{j_{i}}.

Proof. (i). Let r ∈ C(Δ(1)). Since ∑_{j} u_{j}V_{j} = r and w(r) = ∑_{j} u_{j}, we have

∑_{V_{j}∈T} u_{j} ≤ ∑_{j} u_{j} = w(r).

The inequality is an equality if and only if all V_{j_{i}} ∈ T. This is true if and only if Δ(1) contains all the lattice points V_{j_{i}}.

(ii). Let

r_{1} = ∑_{V_{j}∈Δ(1)} u_{j}V_{j}, r_{2} = ∑_{V_{j}∈Δ(2)−H} u_{j}V_{j}.

Then, r = r_{1} + r_{2} and w(r_{1}) = ∑_{V_{j}∈T} u_{j}. If Δ(2) does not contain all of the lattice points V_{j_{i}}, then w(r_{1}) > 0 and r_{1} lies on the other side of Δ(2) with respect to H. This implies that the line segment from r_{1} to r intersects the hyperplane H at a unique point r′_{1} (≠ r_{1}). That is, there is a unique positive number 0 < λ < 1 such that λr_{1} + (1 − λ)r = r′_{1} ∈ H. Thus,

r = λr + (1 − λ)r = λ(r_{1} + r_{2}) + (1 − λ)r = (λr_{1} + (1 − λ)r) + λr_{2} = r′_{1} + λr_{2}.

Since w(r) = w(r_{1}) + w(r_{2}) = w(r′_{1}) + λw(r_{2}), where 0 < λ < 1, we deduce that w(r′_{1}) > w(r_{1}). This shows that w(r_{1}) = ∑_{V_{j}∈T} u_{j} is not the maximun value m(T, Δ; r). This proves (ii). One checks that (iii) follows from (i) and (ii). The lemma is proved.

By analogy with Lemma 3.3, it follows from Lemma 3.5 that we have

Lemma 3.6. (i). Let r be a rational point in the cone C(Δ). Then, m(T, Δ; pr) ≥ pm(T, Δ; r) for all positive rational numbers p. (ii). Let r_{1} and r_{2} be two rational points in the cone C(Δ). Then m(T, Δ; r_{1} + r_{2}) ≥ m(T, Δ; r_{1}) + m(T, Δ; r_{2}). If the equality holds, then both r_{1} and r_{2} lie on C(Δ(1)) or C(Δ(2)).

Proof. One can easily checks (i). We now prove (ii). Let

r_{1} = u_{1}V_{1} + ⋯ + u_{J}V_{J}, ∑_{V_{j}∈T} u_{j} = m(T, Δ; r_{1}),

r_{2} = w_{1}V_{1} + ⋯ + w_{J}V_{J}, ∑_{V_{j}∈T} w_{j} = m(T, Δ; r_{2}),

Then,

r_{1} + r_{2} = (u_{1} + w_{1})V_{1} + ⋯ + (u_{J} + w_{J})V_{J}.  (3.4)
m(T, Δ; r_{1}) + m(T, Δ; r_{2}) = ∑_{V_{j}∈T}(u_{j} + w_{j}) ≤ m(T, Δ; r_{1} + r_{2}).  (3.5)

Let V_{j_{i}} be the lattice points with non-zero coefficients in (3.4). If r_{1} + r_{2} ∈ C(Δ(1)), Lemma 3.5 shows that the inequality in (3.5) is an equality if and only if the V_{j_{i}} are contained in Δ(1). This implies that both r_{1} and r_{2} are contained in the cone C(Δ(1)). If r_{1} + r_{2} ∈ C(Δ(2)) and the inequality in (3.5) is an equality, then Lemma 3.5 shows that the V_{j_{i}} are contained in Δ(2). This implies that both r_{1} and r_{2} are contained in the cone C(Δ(2)). The lemma is proved.

§4. Newton Polygons and Degree Polygons

The purpose of this section is to define the Newton polygon and the degree polygon of a subcone of C(Δ). First, we define the Newton polygon. Let Δ be an arbitrary n-dimensional integral convex polyhedron in R^{n} containing the origin. Let Σ be a cone contained in C(Δ) (not necessarily open or closed). Define a function on non-negative integers as follows:

W_{Σ}(k) = card\{r ∈ Z^{n} ∩ Σ | w(r) = \frac{k}{D(Δ)}\},  (4.1)

Let P (Σ) be the polygon in R^{2} with vertices (0, 0) and

(∑_{k=0}^{m} W_{Σ}(k), \frac{1}{D(Δ)} ∑_{k=0}^{m} kW_{Σ}(k)), m = 0, 1, 2, · · · .  (4.2)

For convenience, we shall call the vertex in (4.2) the m^{th} vertex in P (Σ). Note that the m^{th} vertex may be equal to the (m + 1)^{th} vertex, because it may happen that W_{Σ}(m) = 0. Recall that A_{1}(f) = (a_{s,r}(f)) is the semilinear Frobenius matrix defined in (2.16). We define A_{1}(Σ) to be the submatrix (a_{s,r}(f)) with r and s running through the cone Σ. From the block form (2.25), it follows that

Proposition 4.1. The Newton polygon of det(I − tA_{1}(Σ)) lies above P (Σ).

Next, we define the degree polygon. We assume that the origin 0 is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin. Let the V_{j} be lattice points on Δ_{1} which include all vertices of Δ_{1}. Let T be a subset of the set {V_{j}}. Define a function on non-negative integers as follows:

Q_{Σ}(T, k) = (p − 1) ∑_{w(r)=k,r∈Z^{n}∩Σ} m(T, Δ; r).  (4.3)

Let Q(T, Σ) be the polygon in R^{2} which is the graph of the piecewise linear function with vertices (0, 0) and

(∑_{k=0}^{m} W_{Σ}(k), ∑_{k=0}^{m} Q_{Σ}(T, k)), m = 0, 1, 2, · · · .  (4.4)

Similarly, we shall call the vertex in (4.4) the m^{th} vertex in Q(T, Σ). Even though P (Σ) is a convex polygon, we do not know whether or not Q(T, Σ) is convex.

Recall that f(x) = ∑_{j=1}^{J} a_{j}x^{V_{j}} and T is a subset of the V_{j}. For any polynomial F in the variables a_{j}, we define m(T, F) to be the total degree of F in those variables a_{j} with V_{j} ∈ T. If F = 0, define m(T, F) = −∞. Let F_{r} be the polynomial defined by (2.9) and (2.16). Then, one checks that

m(T, F_{r}) ≤ m(T, Δ; r).

Definition 4.2. Let P (Σ, x) (resp. Q(T, Σ, x)) be the piecewise linear function whose graph is P (Σ) (resp. Q(T, Σ)). By Proposition 4.1, we can write

det(I − tA_{1}(Σ)) = ∑_{k=0}^{∞} p^{P (Σ,k)}G(Σ, k)t^{k},

where G(Σ, k) is a power series in the a_{j} with p-adic integral coefficients. The reduction Ḡ(Σ, k) = G(Σ, k) (mod π) is a polynomial in the a_{j} defined over F̄_{p}. The degree polygon of det(I − tA_{1}(Σ)) with respect to the Newton polygon P (Σ) is defined to be the graph of the piecewise linear function with the vertices

(∑_{k=0}^{m} W_{Σ}(k), m(T, Ḡ(Σ, ∑_{k=0}^{m} W_{Σ}(k)))) (m = 0, 1, ⋯, ∞),

where the second coordinate is the degree of Ḡ(Σ, ∑_{k=0}^{m} W_{Σ}(k)) in terms of T.

Proposition 4.3. (i). The degree polygon of det(I − tA_{1}(Σ)) lies below Q(T, Σ). (ii). If for all rational points r in the cone Σ and all non-negative solutions of the equation ∑_{j=1}^{J} u_{j}V_{j} = r, we always have ∑_{V_{j}∈T} u_{j} = m(T, Δ; r), then the degree polygon of det(I − tA_{1}(Σ)) coincides with its upper bound Q(T, Σ) if and only if the generic Newton polygon (associated with the V_{j}) of det(I − tA_{1}(Σ)) coincides with its lower bound P (Σ).

Proof. Let m be a non-negative integer. By the block form of the matrix A_{1}(Σ), the determinant expansion of a matrix and Lemma 3.6, we deduce that

m(T, Ḡ(Σ, ∑_{k=0}^{m} W_{Σ}(k))) ≤ max_{σ} ∑_{w(r)≤m} m(T, F_{pr−σ(r)})

≤ max_{σ} ∑_{w(r)≤m} m(T, Δ; pr − σ(r))

≤ max_{σ} ∑_{w(r)≤m}(pm(T, Δ; r) − m(T, Δ; σ(r)))

≤ (p − 1) ∑_{w(r)≤m} m(T, Δ; r)

= ∑_{k=0}^{m} Q_{Σ}(T, k),

where r ∈ Z^{n} ∩ Σ and σ runs through the permutations on ∑_{k=0}^{m} W_{Σ}(k) letters. This proves (i).

Under the assumption of (ii), the polynomial F_{r} is homogeneous of degree m(T, Δ; r) in terms of the variables corresponding to T. Furthermore, the identity

m(T, Δ; r_{1} + r_{2}) = m(T, Δ; r_{1}) + m(T, Δ; r_{2})

is true for all rational points r_{1} and r_{2} of the cone Σ. This identity implies that the polynomial Ḡ(Σ, ∑_{k=0}^{m} W_{Σ}(k)) is also homogeneous in terms of the variables corresponding to T. Thus, this polynomial either vanishes or has the desired degree ∑_{k=0}^{m} Q_{Σ}(T, k). The lemma is proved.

Proposition 4.4. If the degree polygon of det(I − tA_{1}(Σ)) coincides with Q(T, Σ) at the m^{th} vertex, then the Newton polygon of det(I − tA_{1}(Σ)) coincides with P (Σ) at the m^{th} vertex. If T contains all the V_{j}, then the converse is also true.

Proposition 4.4 follows directly from the definition of the degree polygon and (ii) of Proposition 4.3. Thus, the degree polygon is finer than the Newton polygon. Grothendieck’s specialization theorem says that the Newton polygon goes up under specializations. The following is an analogue. This proposition follows from the fact that the degree of a polynomial decreases under specializations.

Proposition 4.5. The degree polygon of det(I − tA_{1}(Σ)) defined with respect to the Newton polygon P (Σ) goes down under specializations, where a specialization means a specialization of the affine space whose coordinates are the a_{j} with V_{j} ∉ T.

§5. Boundary Decompositions and Facial Decompositions

Let Δ be an n-dimensional integral convex polyhedron in R^{n} containing the origin. In this section, we give two preliminary decomposition theorems which follow from the properties of the weight function w(r) that were described in Lemma 2.1.

First, we consider the boundary (or interior) decomposition theorem. Let B(Δ) be the unique interior decomposition of the cone C(Δ) into a union of disjoint relatively open cones. Its elements are the interiors of those closed faces in C(Δ) which contain the origin. Thus, the interior cone of C(Δ) is the unique element in B(Δ) of dimension n. The origin itself (if it is a vertex) is the unique element in B(Δ) of dimension 0. For Σ ∈ B(Δ), recall that A_{1}(Σ) is the “Σ” piece (a_{s,r}(f)) in A_{1}(f) such that r and s run through the cone Σ. Let A_{1}(Σ, f_{Σ}) be the “interior piece” of the Frobenius matrix A_{1}(f_{Σ}), where f_{Σ} is the restriction of f to the closure of Σ. The boundary decomposition theorem is the following

Theorem 5.1. We have

det(I − tA_{1}(f)) = ∏_{Σ∈B(Δ)} det(I − tA_{1}(Σ, f_{Σ})).  (5.1)

In particular, the Newton polygon of det(I −tA_{1}(f)) coincides with its lower bound P (Δ) if and only if the Newton polygon of det(I − tA_{1}(Σ, f_{Σ})) coincides with its lower bound P (Σ) for all Σ ∈ B(Δ).

Proof. First, we claim that A_{1}(Σ) coincides with the matrix A_{1}(Σ, f_{Σ}). To prove the claim, let r and s be two lattice points of Σ. We need to prove that F_{ps−r}(f) = F_{ps−r}(f_{Σ}). Let u_{j} be non-negative integers satisfying ∑_{j} u_{j}V_{j} = ps − r. Without loss of generality, we assume that u_{j} > 0 for all j ≤ k and u_{j} = 0 for all j > k. From the equation ps = r + ∑_{j} u_{j}V_{j}, we deduce that s is in the interior of the cone spanned by the V_{j} (all j ≤ k) and r. Because of the nature of the interior decomposition, we conclude that the V_{j} (all j ≤ k) and r are in the closure of the cone Σ. This implies that F_{ps−r}(f) = F_{ps−r}(f_{Σ}). The claim is proved.

Now, fix an ordering of B(Δ) = {Σ_{0}, ⋯, Σ_{h}} such that dim(Σ_{i}) ≤ dim(Σ_{i+1}). Let i < j, s ∈ Σ_{i} and r ∈ Σ_{j}. We prove that ps − r lies outside of C(Δ), that is, w(ps − r) = +∞ and F_{ps−r}(f) = 0. Otherwise, suppose that ps − r lies in C(Δ). Let ps − r ∈ Σ_{k}. From the identity ps = r + (ps − r), we deduce that s lies in the interior of the cone spanned by Σ_{j} and Σ_{k}. This means that dimΣ_{i} ≥ dimΣ_{j} with equality holding only if ps − r ∈ Σ̄_{j}. Since dimΣ_{i} ≤ dimΣ_{j}, we must have ps − r ∈ Σ̄_{j}. From the identity ps = r + (ps − r) again, we conclude that s ∈ Σ_{j}. This shows that Σ_{i} and Σ_{j} are not disjoint, a contradiction. Thus, ps − r lies outside of C(Δ).

Let B_{ij} (0 ≤ i, j ≤ h) be the nuclear submatrix of A_{1}(f) consisting of all (a_{s,r}(f)) with s ∈ Σ_{i} and r ∈ Σ_{j}. Then, under a permutation of the orthonormal basis, we have the block form

A_{1}(f) = ( \begin{matrix}B_{00} & B_{01} & … & B_{0h} \\ B_{10} & B_{11} & … & B_{1h} \\ ⋮ & ⋮ & ... & ⋮ \\ B_{h0} & B_{h1} & … & B_{hh} \end{matrix} ) .  (5.2)

In previous paragraph, we showed that B_{ij} = 0 for i < j. Thus, the block form (5.2) is lower triangular. The theorem follows because the claim shows that B_{ii} = A_{1}(Σ_{i}) = A_{1}(Σ_{i}, f_{Σ_{i}}).

Corollary 5.2. Let Σ ∈ B(Δ). If the Newton polygon of det(I −tA_{1}(f)) coincides with P (Δ), then the Newton polygon of det(I − tA_{1}(f_{Σ})) coincides with its lower bound P (Σ̄).

Proof. Applying (5.1) to f_{Σ}, we see that det(I − tA_{1}(f_{Σ})) is a product of certain factors in (5.1). The corollary follows.

Corollary 5.3. Let d_{1}, ⋯, d_{n} be positive integers. Let D be the least common multiple of the d_{i}. Let f be the general polynomial whose Newton polyhedron is the simplex generated by the origin and the vertices (d_{1}, 0, ⋯, 0), ⋯, (0. · · · , d_{n}). Then the generic Newton polygon of the L-function L^{∗}(f, t)^{(−1)^{n−1}} coincides with its lower bound if and only if p ≡ 1 (modD).

Proof. The sufficiency of this corollary follows from the computation in [15]. Because for the diagonal form f(x) = ∑_{i=1}^{n} x_{i}^{d_{i}}, the Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides with its lower bound for all p ≡ 1 (mod D). If n = 1, the Stickelberger theorem implies that the Newton polygon for f(x) = x^{d_{1}} coincides with its lower bound if and only if p ≡ 1 (mod d_{1}). Thus, the necessity is true for n = 1. If n > 1, the necessity follows from Corollary 5.2 and the case n = 1.

Remark. The referee pointed out that the necessity in general is also implied (at least in many cases) by ramification.

To describe the facial decomposition theorem obtained in [15], let F (Δ) be the set of relatively open cones generated by the relatively open faces in Δ of dimension at most (n − 1) such that the closure of these faces does not contain the origin. The origin itself is considered as an element of F (Δ). Clearly, C(Δ) is the disjoint union of the relatively open cones in F (Δ). For simplicity, we call F (Δ) the open facial decomposition of the cone C(Δ) defined with respect to the origin. The intersection S(Δ) of F (Δ) with Δ gives a decomposition of Δ. This decomposition S(Δ) is called the open facial decomposition of Δ with respect to the origin. If V is any lattice point other than the origin, we can similarly define an open facial decomposition S(Δ, V) of Δ with respect to V. We shall use this construction in next section.

For each Σ ∈ F (Δ), recall that A_{1}(Σ, f_{Σ}) is the “interior piece” of the Frobenius matrix A_{1}(f_{Σ}), where f_{Σ} is the restriction of f to the closure of Σ. Note that there is a difference between A_{1}(Σ, f_{Σ}) and A_{1}(Σ) defined in the last section. The latter matrix is defined in terms of f, while the former is defined in terms of f_{Σ}. We first prove the following open facial decomposition theorem. The closed facial decomposition is proved in Theorem 5.5.

Theorem 5.4. For a non-negative integer m, the Newton polygon of det(I −tA_{1}(f)) coincides with its lower bound at the m^{th} vertex if and only if the Newton polygon of det(I − tA_{1}(Σ, f_{Σ})) coincides with its lower bound at the m^{th} vertex for all Σ ∈ F (Δ).

Proof. The proof is similar to the proof of Theorem 5.1. First, we claim that if Σ ∈ F (Δ) and r belongs to the closure of Σ, then

ord_{p}(F_{r}(f) − F_{r}(f_{Σ})) > \frac{w(r)}{p − 1},  (5.3)

where the right side is the trivial lower bound of ord_{p}(F_{r}(f)) (see (2.13)). To prove the claim, let u_{j} be non-negative integers satisfying ∑_{j} u_{j}V_{j} = r. Without loss of generality, we assume that u_{j} > 0 for all j ≤ k and u_{j} = 0 for all j > k. If now ∑_{j} u_{j} = w(r), then part (c) of Lemma 2.1 shows that the V_{j} (all j ≤ k) are cofacial with respect to the origin. Since the V_{j} (all j ≤ k) also span r, it follows that the V_{j} (all j ≤ k) lies in the closure of Σ. This shows that if a monomial of p-adic order w(r)/(p − 1) appears in F_{r}(f), then this monomial also appears in F_{r}(Σ). The claim is proved.

Now, fix an ordering of F (Δ) = {Σ_{0}, ⋯, Σ_{h}} such that dim(Σ_{i}) ≤ dim(Σ_{i+1}). Let i < j, s ∈ Σ_{i} and r ∈ Σ_{j}. We prove that r and ps − r are not cofacial with respect to the origin, that is, w(r) + w(ps − r) > pw(s) (by Lemma 2.1). Otherwise, suppose that r and ps − r are cofacial with respect to the origin. From the identity ps = r + (ps − r), we deduce that r, s and ps − r are cofacial with respect to the origin. This shows that Σ_{i} and Σ_{j} are two elements in the boundary decomposition of certain piece of C(Δ). Using the identity ps = r+(ps−r) and the same argument as in the second paragraph of the proof of Theorem 5.1, we get a contradiction. This proves that r and ps − r are not cofacial with respect to the origin and w(r) + w(ps − r) > pw(s).

Let B_{ij} (0 ≤ i, j ≤ h) be the nuclear submatrix of A_{1}(f) consisting of all (a_{s,r}(f)) with s ∈ Σ_{i} and r ∈ Σ_{j}. Then we also have the block form (5.2) for A_{1}(f). In previous paragraph, we showed that this block form is in some sense lower triangular. Since B_{ii} = A_{1}(Σ_{i}), the triangular property shows that the Newton polygon of det(I −tA_{1}(f)) coincides with its lower bound at the m^{th} vertex if and only if the Newton polygon of det(I − tA_{1}(Σ)) coincides with its lower bound at the m^{th} vertex for all Σ ∈ F (Δ). Although the matrix A_{1}(Σ) is very different from the desired matrix A_{1}(Σ, f_{Σ}), the Newton polygon of det(I − tA_{1}(Σ)) and the Newton polygon of det(I − tA_{1}(Σ, f_{Σ})) have the same lower bound expressed in terms of W_{Σ}(k).

To finish the proof, we need to show that if we replace the matrix A_{1}(Σ) defined in terms of f by the matrix A_{1}(Σ, f_{Σ}) defined in terms of f_{Σ}, we will not change the property of the coincidence of the Newton polygon with its lower bound. Let r, s ∈ Σ. If ps−r belongs to the closure of Σ, then (5.3) shows that we can replace F_{ps−r}(f) by F_{ps−r}(f_{Σ}). If ps − r does not belong to the closure of Σ, then ps − r and r are not cofacial (one can apply the argument of the second paragraph). Then, the p-order of a_{s,r}(f) = π^{w(r)−w(s)}F_{ps−r} is strictly larger than the expected number w(s), while a_{s,r}(f_{Σ}) = 0. In this case, we can also replace F_{ps−r}(f) by F_{ps−r}(f_{Σ}). The theorem is proved.

The following is the closed facial decomposition theorem.

Theorem 5.5. For a non-negative integer m, the Newton polygon of det(I −tA_{1}(f)) coincides with its lower bound P (Δ) at the m^{th} vertex if and only if the Newton polygon of det(I − tA_{1}(f_{Σ})) coincides with its lower bound P (Σ̄) at the m^{th} vertex for all n-dimensional Σ ∈ F (Δ). Note that A_{1}(f_{Σ}) is the semilinear Frobenius matrix for the restriction f_{Σ} of f to the closure of Σ.

Proof. Let B(Σ̄) be the open boundary decomposition of Σ̄ (or Σ̄ ∩ Δ). Then, one checks that

F (Δ) = ⋃_{Σ∈F (Δ), dim(Σ)=n} B(Σ̄).

Theorem 5.5 then follows from Theorem 5.1 and Theorem 5.4.

Now, Theorem 8 follows immediately from Theorem 5.5 and the theorem of Adolphson and Sperber. For Kloosterman type sums, one can use Theorem 8 to completely classify all prime numbers p such that the Newton polygon coincides with its lower bound. This would improve the Newton polygon results obtained by Sperber [13] and Carpentier [4].

§6. Star Decompositions

The purpose of this section is to establish a star decomposition theorem for Δ. The results will be weaker. But they are strong enough for the generic Newton polygon. Since we are only concerned with the non-emptiness of the Hasse domain H_{p}(Δ), Theorem 5.5 shows that we may assume that the origin 0 is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin. Let V_{1}, ⋯, V_{J} be J fixed lattice points on the face Δ_{1} which include all vertices of Δ_{1}. Then Δ is generated by 0 and the V_{j}. Let C(Δ) be the cone generated by Δ. Let f be the general Laurent polynomial f = ∑_{j} a_{j}x^{V_{j}}. Clearly, the Newton polyhedron of f is Δ. In this section, we take T = V_{1} (not necessarily a vertex). Thus, the degree polygon is defined in terms of V_{1}.

We first describe the open star decomposition of Δ_{1} with respect to V_{1}. Let S(Δ_{1}, V_{1}) be the open facial decomposition (see §5) of the (n − 1)-dimensional Δ_{1} defined with respect to V_{1}. The vertex V_{1} is an element of S(Δ_{1}, V_{1}). For σ ∈ S(Δ_{1}, V_{1}), define Σ(σ) to be the open cone spanned by σ and the origin. If σ is the vertex V_{1}, then Σ(σ) is the open half line spanned by the origin and V_{1}. We always have

dimΣ(σ) = dimσ + 1.

Let F (Δ, V_{1}) be the set consisting of the origin and all Σ(σ), where σ runs through S(Δ_{1}, V_{1}). Then the closed cone C(Δ) is the disjoint union of the open cones Σ in F (Δ, V_{1}). In particular, we have

W (k) = ∑_{Σ∈F (Δ,V_{1})} W_{Σ}(k).

We call F (Δ, V_{1}) the open star decomposition of C(Δ) with respect to V_{1}. The open facial decomposition in §5 is simply the open star decomposition with respect to the origin.

Unless otherwise stated, in this section we shall use σ to denote an element of S(Δ_{1}, V_{1}) and Σ to denote an element of F (Δ, V_{1}). Let f_{Σ} be the restriction of the Laurent polynomial f to the closed cone spanned by Σ. Then f_{Σ} is a Laurent polynomial whose Newton polyhedron is the intersection of Δ and the closure of Σ. Dwork’s theory as described in §2 applies to f_{Σ}. Let A_{1}(Σ, f_{Σ}) be the “interior piece” of the “closed” matrix A_{1}(f_{Σ}) = (a_{s,r}(f_{Σ})) consisting of all a_{s,r}(f_{Σ}) with s and r running through Σ.

The following proposition follows from the block form of A_{1}(Σ, f_{Σ}).

Proposition 6.1. The Newton polygon of det(I − tA_{1}(Σ, f_{Σ})) computed with respect to p lies above P (Σ). The degree polygon of det(I − tA_{1}(Σ, f_{Σ})) defined with respect to P (Σ) lies below Q(V_{1}, Σ).

The open star decomposition theorem is

Theorem 6.2. For any non-negative integer m, the degree polygon of det(I −tA_{1}(f)) coincides with Q(V_{1}, Δ) at the m^{th} vertex if and only if the degree polygon of det(I − tA_{1}(Σ, f_{Σ})) coincides with Q(V_{1}, Σ) at the m^{th} vertex for all Σ ∈ F (Δ, V_{1}).

To prove this result, we first prove

Lemma 6.3. Let Σ ∈ F (Δ, V_{1}), r ∈ Σ̄ and f = ∑_{j=1}^{J} a_{j}x^{V_{j}}. Then

m(V_{1}, F_{r}(f) − F_{r}(f_{Σ})) < m(V_{1}, Δ; r).

That is, the total degree of F_{r}(f) − F_{r}(f_{Σ}) in the variable a_{1} is strictly smaller than the expected value m(V_{1}, Δ; r).

Proof. Let u_{1}, ⋯, u_{J} be non-negative integers satisfying r = ∑_{j} u_{j}V_{j}. Without loss of generality, we may assume that u_{1} ≥ 0, u_{j} > 0 for 2 ≤ j ≤ k and u_{j} = 0 for j > k. Assume that u_{1} = m(V_{1}, Δ; r). Lemma 3.2 shows that the V_{j} (2 ≤ j ≤ k) are cofacial with respect to V_{1}. Since r − u_{1}V_{1} = ∑_{j=2}^{k} u_{j}V_{j}, the construction of the open star decomposition shows that the V_{j} (j ≤ k) lie in the closure of the cone Σ. This shows that if a monomial in the a_{j} of degree m(V_{1}, Δ; r) with respect to a_{1} appears in F_{r}(f), then the same monomial also appears in F_{r}(f_{Σ}). Thus, F_{r}(f) and F_{r}(f_{Σ}) have the same initial terms of degree m(V_{1}, Δ; r) with respect to a_{1}. The lemma is proved.

Proof of Theorem 6.2. Fix an ordering of F (Δ, V_{1}) = {Σ_{0}, ⋯, Σ_{h}} such that dim(Σ_{i}) ≤ dim(Σ_{i+1}). Let i < j, s ∈ Σ_{i} and r ∈ Σ_{j}. We claim that r and ps − r are not cofacial with respect to V_{1}. Otherwise, suppose that r and ps − r are cofacial with respect to V_{1}. From the identity ps = r+(ps−r), it follows that r, s and ps−r are all cofacial with respect to V_{1}. Using the identity ps = r + (ps − r) and the same argument as in the second paragraph of the proof of Theorem 5.1, we get a contradiction. The claim is proved. By Lemma 3.3, we then have

m(V_{1}, a_{s,r}(f)) = m(V_{1}, F_{ps−r}) ≤ m(V_{1}, Δ; ps − r) < pm(V_{1}, Δ; s) − m(V_{1}, Δ; r).  (6.1)

Let B_{ij} (0 ≤ i, j ≤ h) be the nuclear submatrix of A_{1}(f) consisting of all (a_{s,r}(f)) with s ∈ Σ_{i} and r ∈ Σ_{j}. The Newton polygon of the entire function det(I−tB_{ii}) lies above P (Σ_{i}). Furthermore, under a permutation of orthonormal basis,

A_{1}(f) = ( \begin{matrix}B_{00} & B_{01} & … & B_{0h} \\ B_{10} & B_{11} & … & B_{1h} \\ ⋮ & ⋮ & ... & ⋮ \\ B_{h0} & B_{h1} & … & B_{hh} \end{matrix} ) .  (6.2)

If a_{s,r}(f) is an element in B_{ij} with i < j, then (6.1) shows that the degree m(V_{1}, a_{s,r}(f)) of the polynomial a_{s,r}(f) in the variable a_{1} is strictly smaller than the expected value pm(V_{1}, Δ; s) − m(V_{1}, Δ; r). This means that the above block form for A_{1}(f) is, in some sense, lower triangular with respect to the degree of a_{1}. By induction, we deduce that

det(I − tA_{1}(f)) = ∏_{i=0}^{h} det(I − tB_{ii}) + ∑_{k=0}^{∞} p^{P (Δ,k)}G(k)t^{k},  (6.3)

where G(k) is a power series in the a_{j} with p-adic integral coefficients such that the reduction Ḡ(k) = G(k) (mod π) is a polynomial over F̄_{p} whose degree in a_{1} is strictly smaller than their upper bound (i.e., the number Q(V_{1}, Δ, k), see the notation in Definition 4.2). Thus, the degree polygon of det(I − tA_{1}(f)) defined with respect to P (Δ) coincides with Q(V_{1}, Δ) at the m^{th} vertex if and only if the degree polygon of the first term on the right side of (6.3) (defined with respect to P (Δ)) coincides with Q(V_{1}, Δ) at the m^{th} vertex. One further shows that the latter is true if and only if the degree polygon of det(I − tA_{1}(Σ)) defined with respect to P (Σ) coincides with Q(V_{1}, Σ) at the m^{th} vertex for all Σ ∈ F (Δ, V_{1}). Although the matrix A_{1}(Σ) is very different from the desired matrix A_{1}(Σ, f_{Σ}), the degree polygon of det(I − tA_{1}(Σ)) and the degree polygon of det(I − tA_{1}(Σ, f_{Σ})) have the same upper bound Q(V_{1}, Σ).

To finish the proof, we need to show that if we replace the matrix A_{1}(Σ) defined in terms of f by the matrix A_{1}(Σ, f_{Σ}) defined in terms of f_{Σ}, we will not change the property of the coincidence of the degree polygon with its upper bound. Let r, s ∈ Σ. If ps − r belongs to the closure of Σ, Lemma 6.3 shows that we can replace F_{ps−r}(f) by F_{ps−r}(f_{Σ}). If ps − r does not belong to the closure of Σ, then ps − r and r are not cofacial with respect to V_{1} (one can apply the argument of the first paragraph). Then, the degree in a_{1} of F_{ps−r}(f) is strictly smaller than the expected number pm(V_{1}, Δ; s) − m(V_{1}, Δ; r), while F_{ps−r}(f_{Σ}) = 0. In this case, we can also replace F_{ps−r}(f) by F_{ps−r}(f_{Σ}). The theorem is proved.

Applying Theorem 6.2 and the boundary decomposition theorem (Theorem 5.1), we deduce the following closed star decomposition theorem.

Theorem 6.4. The degree polygon of det(I − tA_{1}(f)) coincides with Q(V_{1}, Δ) at the m^{th} vertex if and only if the degree polygon of det(I − tA_{1}(f_{Σ})) defined with respect to P (Σ̄) coincides with Q(V_{1}, Σ̄) at the m^{th} vertex for all n-dimensional Σ ∈ F (Δ, V_{1}).

Proof. The proof is similar to the proof of Theorem 5.5. Let Σ ∈ F (Δ, V_{1}) be an n-dimensional cone. Let B(Σ̄) be the open boundary decomposition of the closed cone Σ̄ (or the closed polyhedron Σ̄ ∩ Δ). Then, one checks that

F (Δ, V_{1}) = ⋃_{Σ∈F (Δ,V_{1}), dim(Σ)=n} B(Σ̄).

Theorem 6.4 then follows from Theorem 5.1 and Theorem 6.2.

We are now in a position to prove that the degree polygon coincides with its upper bound if p lies in a certain residue class.

Theorem 6.5. Let Δ have only one face of codimension 1 which does not contain the origin. Let V_{1}, ⋯, V_{J} be J lattice points on Δ which include all non-zero vertices of Δ. Let f be the general Laurent polynomial ∑_{j} a_{j}x^{V_{j}}. Let V_{1} be a lattice point (not necessarily a vertex) on the unique face of codimension 1 not containing the origin. Let m be a fixed non-negative integer. There is an effectively computable positive integer D^{∗}(m) depending only on the V_{j} and m such that if p ≡ 1 (mod D^{∗}(m)), then the degree polygon of det(I − tA_{1}(f)) with respect to V_{1} coincides with its upper bound Q(V_{1}, Δ) at the m^{th} vertex.

Proof. For a general Δ, by applying Theorem 6.4 we may assume that V_{1} is a vertex. The degree polygon is defined with respect to V_{1}. By the specialization theorem (Proposition 4.5), we may further assume that the V_{j} are the non-zero vertices in Δ. Note that under such specializations, the upper bound remains fixed.

We first assume that Δ is a simplex with vertices 0, V_{1}, ⋯, V_{n}. Let D^{∗} be the torsion order of the torsion abelian group Z^{n}/ZV_{1} + ⋯ + ZV_{n}. For r ∈ C(Δ), the equation

∑_{j=1}^{n} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} rational

has exactly one solution and this unique solution will make u_{1} maximal. In the notation of (2.9), F_{r}(f) is either zero or a monomial of degree m(V_{1}, Δ; r) with respect to a_{1}. The assumption of part (ii) of Proposition 4.3 is satisfied. We deduce that if the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound P (Δ) at the m^{th} vertex, then the degree polygon of det(I − tA_{1}(f)) with respect to V_{1} coincides with its upper bound Q(V_{1}, Δ) at the m^{th} vertex. Under the assumption that p ≡ 1 (mod D^{∗}), it is shown in [15] that the generic Newton polygon coincides with its lower bound for all sides. Thus, the theorem is true for a simplex Δ.

In general, we showed that we may assume that the V_{j} are the vertices on the unique face of codimension 1 not containing the origin. We now use induction on the number J of V_{j} on Δ. Assume that the theorem is true for all Δ with fewer than J non-zero vertices V_{j}. We now prove that the theorem is true for any Δ with J non-zero vertices V_{j}. If F (Δ, V_{1}) has at least two elements of dimension n, then Theorem 6.4 gives a non-trivial decomposition. By induction, we conclude that the degree polygon (with respect to the vertex V_{1}) of det(I − tA_{1}(f)) coincides with Q(V_{1}, Δ) at the m-th vertex for all p in a certain residue class p ≡ 1 (mod D^{∗}(m)). If F (Δ, V_{1}) has only one element of dimension n, then Δ has only one closed face of codimension 1 not containing V_{1}. In this case, all solutions of the equation

∑_{j=1}^{J} u_{j}V_{j} = r, u_{j} ≥ 0, u_{j} rational

satisfy w(r) = u_{1} + ⋯ + u_{J}. Thus,

r − w(r)V_{1} = ∑_{j=2}^{J} u_{j}(V_{j} − V_{1}).

Since all the V_{j} (all j ≥ 2) are cofacial with respect to V_{1}, one sees that ∑_{j=2}^{J} u_{j} is the weight of r − w(r)V_{1} with respect to the vertices of {V_{j} − V_{1}}_{j=2}^{J}. Hence, ∑_{j=2}^{J} u_{j} depends only on V_{1} and r but not on the particular u ∈ (Z_{≥0})^{J} satisfying ∑_{j=1}^{J} u_{j}V_{j} = r. Thus, u_{1} depends only on V_{1} and r. This proves that any solution (if it exists) will make u_{1} maximal. Thus, the assumption in part (ii) of Proposition 4.3 is satisfied. We deduce that that if the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound P (Δ) at the m^{th} vertex, then the degree polygon of det(I − tA_{1}(f)) with respect to V_{1} coincides with its upper bound Q(V_{1}, Δ) at the m^{th} vertex.

To finish the proof, choose a vertex, say V_{2}, such that F (Δ, V_{2}) has at least two elements of dimension n (if such a V_{2} does not exist, then Δ is already simplicial and we are done). In this way, Theorem 6.4 gives a non-trivial decomposition with respect to V_{2}. By induction we conclude that for all primes in the residue class p ≡ 1 (mod D^{∗}(m)), the degree polygon (and hence the generic Newton polygon) with respect to V_{2} of det(I −tA_{1}(f)) coincides with Q(V_{2}, Δ) (resp. P (Δ)) at the m^{th} vertex. By the discussion in the previous paragraph, this implies that for all such p, the degree polygon with respect to V_{1} of det(I − tA_{1}(f)) also coincides with Q(V_{1}, Δ) at the m^{th} vertex. The theorem is proved.

Remark. Let the V_{j} be the non-zero vertices in Δ. Let f be the smallest family of Laurent polynomials with Δ(f) = Δ, i.e., f(x) = ∑_{i} a_{j}x^{V_{j}}. If Δ is a simplex, one can use the Stickelberger relation to find the smallest positive integer D^{∗} such that whenever p ≡ 1 (mod D^{∗}), the generic Newton polygon of L^{∗}(f, t)^{(−1)^{n−1}} coincides with its lower bound (see the beginning of §1). For a general Δ, the proof of Theorem 6.5 shows that we can still find the smallest positive integer D^{∗} such that whenever p ≡ 1 (mod D^{∗}), the degree polygon of det(I − tA_{1}(f)) with respect to each of the vertices V_{i} coincides with its upper bound. This means that the leading form with respect to each variable a_{i} is non-zero for all of the Hasse polynomials h_{p}(Δ, m). Thus, the D^{∗} so obtained is best possible in the stronger sense that the degree polygon with respect to each of the vertices V_{i} coincides with its upper bound.

Proof of Theorem 3. Let Δ be an n-dimensional integral convex polyhedron in R^{n} containing the origin. The theorem of Adolphson-Sperber shows that L^{∗}(f, t)^{(−1)^{n−1}} is a polynomial of degree n!V(Δ) for general f and large primes p. By (2.21) and Corollary 2.5, it suffices to prove that there is a positive integer D^{∗} such that if p ≡ 1 (mod D^{∗}) and p is large, then the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound P (Δ) for the first nD sides. Theorem 5.5 implies that we may assume that Δ has only one face of codimension 1 not containing the origin. Then we can apply Theorem 6.5 and take D^{∗} to be the least common multiple of the D^{∗}(m) (0 ≤ m ≤ nD). Theorem 3 is proved.

7. Hyperplane Decompositions

The decomposition theorem in §6 is based on a cut of Δ with respect to a lattice point V_{1}. One can consider other types of cuts with respect to several lattice points. In this section, we give a decomposition theorem for hyperplane cuts of Δ. This is then used to prove that Conjecture 1 is true in some special cases.

As in section 6, we assume that the origin 0 is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin. Let V_{1}, ⋯, V_{J} be J fixed lattice points on the face Δ_{1} which include all vertices of Δ_{1}. Then Δ is generated by 0 and the V_{j}. Let f = ∑_{j} a_{j}x^{V_{j}} be the general Laurent polynomial associated with the V_{j}. Let H be a hyperplane passing through the origin such that the intersection of H with Δ is an integral polyhedron of dimension n − 1 whose vertices are contained in the V_{j}. Then H cuts Δ into two pieces Δ(1) and Δ(2). We assume that both Δ(1) and Δ(2) are closed. Then, one sees that Δ is the union of the two pieces Δ(1) and Δ(2). Let T be the set of V_{j} which are contained in Δ(1). Let T_{H} be the intersection of T and H.

For i = 1, 2, let f_{i} be the restriction of the general Laurent polynomial f to Δ(i). Then f_{i} is the generic Laurent polynomial associated with those V_{j} which are contained in Δ(i). The Newton polyhedron of f_{i} is Δ(i). Dwork’s theory as described in §2 applies to f_{i}. The following theorem is an analogue of the closed star decomposition theorem.

Theorem 7.1. Let m be a non-negative integer. The degree polygon of det(I − tA_{1}(f)) defined with respect to T coincides with Q(T, Δ) at the m^{th} vertex if and only if both the Newton polygon of det(I − tA_{1}(f_{1})) coincides with P (C(Δ(1))) at the m^{th} vertex and the degree polygon of det(I − tA_{1}(f_{2})) with respect to T_{H} coincides with Q(T_{H}, Δ(2)) at the m^{th} vertex.

Remark. Since T consists of all the V_{j} which are contained in Δ(1), the Newton polygon of det(I − tA_{1}(f_{1})) coincides with its lower bound P (C(Δ(1))) if and only if the degree polygon of det(I − tA_{1}(f_{1})) with respect to T coincides with its upper bound Q(T, Δ(1)). Thus, Theorem 7.1 is a decomposition theorem about degree polygons.

Proof of Theorem 7.1. Let Σ_{i} (i = 1, 2) be the n-dimensional cone spanned by Δ(i) − H. Let Σ_{0} be the (n − 1)-dimensional cone spanned by Δ ∩ H. Then, the cone C(Δ) is the disjoint union of the Σ_{i} (i = 0, 1, 2). Let 0 ≤ i < j ≤ 2, s ∈ Σ_{i} and r ∈ Σ_{j}. We claim that r and ps − r can not both lie on the cone C(Δ(1)) or C(Δ(2)). If i = 0, it is clear that ps − r will not lie on the closure of Σ_{j} (while r does). If i = 1, then ps − r will not lie on the closure of Σ_{j} (while r does). The claim is proved. Lemma 3.6 shows that

m(T, a_{s,r}(f)) = m(T, F_{ps−r}) ≤ m(T, Δ; ps − r) < pm(T, Δ; s) − m(T, Δ; r).  (7.1)

Let B_{ij} (0 ≤ i, j ≤ 2) be the nuclear submatrix of A_{1}(f) consisting of all (a_{s,r}(f)) with s ∈ Σ_{i} and r ∈ Σ_{j}. The Newton polygon of the entire function det(I−tB_{ii}) lies above P (Σ_{i}). Furthermore, under a permutation of orthonormal basis,

A_{1}(f) = ( \begin{matrix}B_{00} & B_{01} & B_{02} \\ B_{10} & B_{11} & B_{12} \\ B_{20} & B_{21} & B_{22} \end{matrix} ) .  (7.2)

If a_{s,r}(f) is an element in B_{ij} with i < j, then (7.1) shows that the degree m(T, a_{s,r}(f)) of the polynomial a_{s,r}(f) in the variables corresponding to T is strictly smaller than the expected value pm(T, Δ; s) − m(T, Δ; r). This means that the above block form for A_{1}(f) is, in some sense, lower triangular with respect to the degree defined in terms of T. We deduce that

det(I − tA_{1}(f)) = ∏_{i=0}^{2} det(I − tB_{ii}) + ∑_{k=0}^{∞} p^{P (Δ,k)}G(k)t^{k},  (7.3)

where G(k) is a power series in the a_{j} with p-adic integral coefficients such that the reduction Ḡ(k) = G(k) (mod π) is a polynomial over F̄_{p} whose degree in T is strictly smaller than its upper bound (i.e., the number Q(T, Δ, k), see the notation in Definition 4.2) . Thus, the degree polygon of det(I − tA_{1}(f)) defined with respect to P (Δ) coincides with Q(T, Δ) at the m^{th} vertex if and only if the degree polygon of the first term on the right side of (7.3) (defined with respect to P (Δ)) coincides with Q(T, Δ) at the m^{th} vertex. One further shows that the latter is true if and only if the degree polygon of det(I − tB_{ii}) defined with respect to P (Σ_{i}) coincides with Q(T, Σ_{i}) at the m^{th} vertex for all 0 ≤ i ≤ 2. Although the matrix A_{1}(Σ_{i}) is very different from the desired matrix A_{1}(Σ_{i}, f_{Σ_{i}}), the degree polygon of det(I −tA_{1}(Σ_{i})) and the degree polygon of det(I −tA_{1}(Σ_{i}, f_{Σ_{i}})) have the same upper bound Q(T, Σ_{i}).

Now, we need to show that if we replace the matrix A_{1}(Σ_{i}) defined in terms of f by the matrix A_{1}(Σ_{i}, f_{Σ_{i}}) defined in terms of f_{Σ_{i}}, we will not change the property of the coincidence of the degree polygon with its upper bound. Let r, s ∈ Σ_{i}. If ps − r belongs to the closure of Σ_{i}, Lemma 3.5 shows that the degree of F_{ps−r}(f) − F_{ps−r}(f_{Σ_{i}}) in T is strictly smaller than the expected degree m(T, Δ; ps − r). Thus, we can replace F_{ps−r}(f) by F_{ps−r}(f_{Σ_{i}}). If ps − r does not belong to the closure of Σ_{i}, then Lemma 3.6 shows that the degree of F_{ps−r}(f) in T is strictly smaller than the expected degree pm(T, Δ; s) − m(T, Δ; r), while F_{ps−r}(f_{Σ_{i}}) = 0. In this case, we can also replace F_{ps−r}(f) by F_{ps−r}(f_{Σ_{i}}). This establish the “open” decomposition that the degree polygon of det(I − tA_{1}(f)) defined with respect to P (Δ) coincides with Q(T, Δ) at the m^{th} vertex if and only if the degree polygon of det(I − tA_{1}(Σ_{i}, f_{Σ_{i}})) defined with respect to P (Σ_{i}) coincides with Q(T, Σ_{i}) at the m^{th} vertex for all 0 ≤ i ≤ 2.

To finish the proof, we apply the boundary decomposition theorem (Theorem 5.1) and deduce that

det(I − tA_{1}(f_{1})) = det(I − tA_{1}(Σ_{0}, f_{Σ_{0}}))det(I − tA_{1}(Σ_{1}, f_{Σ_{1}})).
det(I − tA_{1}(f_{2})) = det(I − tA_{1}(Σ_{0}, f_{Σ_{0}}))det(I − tA_{1}(Σ_{2}, f_{Σ_{2}})).

Thus, the degree polygon of det(I − tA_{1}(Σ_{i}, f_{Σ_{i}})) coincides with Q(T, Σ_{i}) at the m^{th} vertex for all 0 ≤ i ≤ 2 if and only if the degree polygon of det(I − tA_{1}(f_{i})) defined with respect to P (Δ(i)) coincides with Q(T, C(Δ(i))) at the m^{th} vertex for 1 ≤ i ≤ 2. Theorem 7.1 is proved.

By the chain level version of Conjecture 1, we mean the assertion that the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound for all primes p in the residue class p ≡ 1 (mod D(Δ)), where D(Δ) is the denominator of Δ. If L^{∗}(f, t)^{(−1)^{n−1}} is a polynomial of degree n!V(Δ) for some f (this is true if f is non-degenerate), then the chain level version of Conjecture 1 implies Conjecture 1. The chain level version is stronger because it does not exclude small prime numbers. We now use the above theorems to prove that the chain level version of Conjecture 1 is true in various special cases. For simplicity, in what follows Conjecture 1 means the chain level version.

Assume now that Δ in R^{n+1} is the (n + 1)-dimensional integral convex polyhedron generated by the origin and some n-dimensional face Δ_{1} not containing the origin. By a suitable coordinate change (applied to Laurent polynomials) of the form

x_{j} ⟶ ∏_{i=1}^{n+1} x_{i}^{t_{ij}}, T = (t_{ij}) ∈ GL(n + 1, Z),

we may assume that Δ_{1} lies on the hyperplane x_{n+1} = D. Let the V_{j} be the set of all lattice points in Δ_{1}. We first treat the case when Δ_{1} is an integral unit cube.

Lemma 7.2. Let Δ be generated by the origin and V + ∑_{i=1}^{n} ϵ_{i}e_{i}, where V is a lattice point on the hyperplane x_{n+1} = D, the e_{i} are an integral basis for the lattice points in R^{n} and the ϵ_{i} take values among {0, 1} (i.e., Δ_{1} is an integral unit cube of dimension n). Then Conjecture 1 is true.

Proof. Since Δ_{1} is a unit cube, there are no lattice points on Δ_{1} other than the vertices. We use Theorem 6.4 to decompose Δ with respect to the vertex V. In this way, we get n integral polyhedra Δ(i) (i = 1, ⋯, n), where the non-zero vertices of Δ(i) are V and V + e_{i} + ∑_{j≠i}^{n} ϵ_{j}e_{j}. Since each Δ(i) is simplicial with respect to V and there are no lattice points on Δ(i) ∩ Δ_{1} other than the vertices, one checks that the assumption of part (ii) of Proposition 4.3 is satisfied. We deduce that the degree polygon for Δ(i) with respect to V coincides with its upper bound if and only if the Newton polygon for Δ(i) coincides with its lower bound. Thus, Conjecture 1 for Δ is reduced to the Δ(i). Now, for each i we apply Theorem 6.4 to Δ(i) with respect to the vertex V +e_{i}. We obtain (n−1) integral polyhedra Δ(ij) (j ≠ i), where the non-zero vertices of Δ(ij) are V, V + e_{i} and V + e_{i} + e_{j} + ∑_{k≠j,i}^{n} ϵ_{k}e_{k}. Since each Δ(ij) is simplicial with respect to V + e_{i} and there are no lattice points on Δ(ij) ∩ Δ_{1} other than the vertices, the above argument shows that the degree polygon for Δ(ij) with respect to V + e_{i} coincides with its upper bound if and only if the Newton polygon for Δ(ij) coincides with its lower bound. Thus, Conjecture 1 for Δ is reduced to the Δ(ij). Repeating the above procedure and using induction, we see that Conjecture 1 for Δ is reduced to the simplicial Δ(i_{1}i_{2} ⋯ i_{n}), where i_{1} ⋯ i_{n} runs over the permutations of {1, 2, ⋯, n} and the non-zero vertices of Δ(i_{1}i_{2} ⋯ i_{n}) are the column vectors of the matrix

M = (V, V + e_{i_{1}}, V + e_{i_{1}} + e_{i_{2}}, ⋯, V + e_{i_{1}} + ⋯ + e_{i_{n}}).

Clearly, det(M) = D det(e_{i_{1}}, ⋯, e_{i_{n}}). Since the e_{i} are an integral basis for the lattice points in R^{n}, the determinant of M has absolute value D. This proves that Conjecture 1 is true for each Δ(i_{1}i_{2} ⋯ i_{n}), and hence also for Δ.

Next, we treat the case when Δ is a general integral box.

Theorem 7.3. Let Δ be generated by the origin and V +∑_{i=1}^{n} ϵ_{i}d_{i}e_{i}, where the d_{i} are positive integers (thus, Δ_{1} is a box of dimension n). Then Conjecture 1 is true.

Proof. The idea is to use Theorem 7.1 to reduce the theorem to the case treated in Lemma 7.2. We use induction on the number d = ∑_{i} d_{i}. Lemma 7.2 shows that the theorem is true for d = n (all d_{i} = 1). Assume now that d > n. Then one of the d_{i}, say d_{1}, is greater than 1. Let H be the hyperplane generated by the origin and the vertices V + (d_{1} − 1)e_{1} + ∑_{j=2}^{n} ϵ_{j}d_{j}e_{j}. Similarly, let K be the hyperplane generated by the origin and the vertices V + d_{1}e_{1} + ∑_{j=2}^{n} ϵ_{j}d_{j}e_{j} (these vertices are the vertices of a boundary face of Δ_{1}). Then H cuts Δ into two pieces Δ(1) and Δ(2), where the non-zero vertices of Δ(1) are V +ϵ_{1}(d_{1}−1)e_{1}+∑_{i=2}^{n} ϵ_{i}d_{i}e_{i} and the non-zero vertices of Δ(2) are V + (d_{1} − 1 + ϵ_{1})e_{1} + ∑_{i=2}^{n} ϵ_{i}d_{i}e_{i}. To apply Theorem 7.1, we need to prove that the degree polygon for Δ(2) with respect to the lattice points in H coincides with its upper bound. Let V_{1}, ⋯, V_{h} be the lattice points on the unique face of Δ(2) not containing the origin. For each j ≤ h, V_{j} lies on one of the hyperplanes H or K. If V_{j} lies on H, then the e_{1} coordinate of V_{j} − V is d_{1} − 1. If V_{j} lies on K, then the e_{1} coordinate of V_{j} − V is d_{1}. Without loss of generality, we may assume that V_{1}, ⋯, V_{k} lie on H and the rest of the V_{j} (all j ≤ h) lie on K. Thus, T_{H} = {V_{1}, ⋯, V_{k}}.

Let r be a rational point in the cone generated by Δ(2). We may assume that w(r) = 1. Then r lies on the unique face of Δ(2) not containing the origin. We can write

r = V + (d_{1} − 1 + c_{1})e_{1} + ∑_{j=2}^{n} c_{j}d_{j}e_{j}, 0 ≤ c_{j} ≤ 1,

where c_{1} is uniquely determined by r because d_{1} −1+c_{1} is the e_{1} coordinate of r−V. If (u_{1}, ⋯, u_{n}) is a non-negative rational solution of the equation ∑_{j=1}^{h} u_{j}V_{j} = r, then w(r) = ∑_{j=1}^{h} u_{j} = 1 implies that

∑_{j=1}^{h} u_{j}(V_{j} − V) = r − V.

Comparing the e_{1} coordinate, we see that

(∑_{j=1}^{k} u_{j})(d_{1} − 1) + ( ∑_{j=k+1}^{h} u_{j})d_{1} = (d_{1} − 1 + c_{1}).

w(r) = ∑_{j=1}^{k} u_{j} + ∑_{j=k+1}^{h} u_{j} = 1.

This shows that (u_{1} + ⋯ + u_{k}) depends only on c_{1}, and hence only on r. Thus, any solution will make ∑_{i=1}^{k} u_{i} maximal and the assumption of part (ii) of Proposition 4.3 is satisfied. We deduce that if the generic Newton polygon for Δ(2) coincides with its lower bound, then the degree polygon for Δ(2) with respect to the lattice points in T_{H} coincides with its upper bound. By induction, Conjecture 1 is true for Δ(2). Hence, the degree polygon for Δ(2) coincides with its upper bound. By induction, Conjecture 1 is also true for Δ(1). We can then apply Theorem 7.1 to conclude that Conjecture 1 is true for Δ. The theorem is proved.

To treat more general cases, we define an incomplete integral unit cube to be an integral convex polyhehron contained in an integral unit cube. If we intersect an arbitrary Δ_{1} with an integral unit cube, we obtain an incomplete cube but in general it might not be integral (i.e., the vertices of the incomplete cube may not be lattice points).

Theorem 7.4. Let Δ_{1} be an n-dimensional integral convex polyhedron lying on the hyperplane x_{n+1} = D. (i). Assume that the intersections of Δ_{1} with all n-dimensional integral unit cubes in R^{n} (lying on the hyperplane x_{n+1} = D) are integral and assume that Conjecture 1 is true for all of these intersections. Then Conjecture 1 is true for Δ, where Δ is generated by the origin and Δ_{1}. (ii). Assume that {H_{j}} (1 ≤ j ≤ k) is a finite sequence of successive parrallel hyperplanes in the n-dimensional space x_{n+1} = D (thus, there are no lattice points lying strictly between H_{j} and H_{j+1}). Assume that H_{1} and H_{k} do not intersect the interior of Δ_{1}. Assume that this sequence of hyperplances {H_{j}} cut Δ_{1} into finitely many integral polyhedra and assume that Conjecture 1 is true for each of these polyhedra, then Conjecture 1 is true for Δ.

Proof. (i). It is clear that Δ_{1} is contained in some n-dimensional integral box. We can then repeat the proof of Theorem 7.3 and use induction. In this case, we also obtain some incomplete integral unit cubes for which we can use our assumptions.

(ii). The proof is to use Theorem 7.1 and induction on the number k. If k ≤ 2, it is trivial. If k > 2, let H be the space spanned by H_{2} and the origin. Then, H is a hyperplane in the (n + 1)-dimensional space. One sees that H non-trivially cuts Δ into two pieces Δ(1) and Δ(2). Here Δ(2) denotes the piece intesecting H_{1}. Thus, T is the set of lattice points in Δ(1) ∩ Δ_{1} and T_{H} is the the set of lattice points in H ∩ Δ_{1}. By induction and our assumptions, Conjecture 1 is true for both Δ(1) and Δ(2). Since H_{1} does not intersect the interior of Δ(2) and H_{1} is successive and parrallel to H_{2}, the proof of Theorem 7.3 shows that the degree polygon for Δ(2) with respect to T_{H} coincides with its upper bound if and only if the Newton polygon for Δ(2) coincides with its lower bound. The latter is true by our assumption. By induction, the Newton polygon for Δ(1) also coincides with its lower bound. The theorem then follows from Theorem 7.1.

For applications to the zeta functions of affine hypersurfaces of degree d, we need to consider the case that Δ is the (n + 1)-dimensional simplex generated by the origin, V and V + de_{i}. Thus, Δ_{1} is the n-dimensional simplex with vertices V and V + de_{i}.

Theorem 7.5. Let Δ be generated by the origin, V and V + de_{i}, where d is a positive integer and V is a lattice point on the hyperplane x_{n+1} = D. Then Conjecture 1 is true.

Proof. To simplify notations, we may assume that Δ_{1} is the n-dimensional simplex in R^{n} generated by the origin and the vertices de_{i}. We may also assume that the e_{i} are the standard basis. Now, we use the regular subdivision of a simplex as described in [8, p117-119]. Define the cumulative coordinates as follows:

\begin{matrix}y_{1} = & x_{1}, \\ y_{2} = & x_{1} + x_{2}, \\ ⋯ & ⋯ \\ y_{n} = & x_{1} + x_{2} + ⋯ + x_{n}. \end{matrix}

One sees that Δ_{1} is identified with the set of n-tuples of real numbers y_{1}, ⋯, y_{n} such that

0 ≤ y_{1} ≤ y_{2} ≤ ⋯ ≤ y_{n} ≤ d.

Consider the following hyperplanes in R^{n}:

H(i, j; k) : y_{i} − y_{j} = k, where 0 ≤ j < i ≤ n, 0 ≤ k ≤ d.

We claim that these hyperplanes cut Δ_{1} into integral simplexes with determinant 1 (equivalently, with volume 1/n!). To prove the claim, let y_{1}, ⋯, y_{n} be a point in Δ′_{1} = Δ_{1} − ∪_{i,j,k}H(i, j; k). If we put t_{i} = y_{i} − [y_{i}], it is clear that t_{i} ≠ t_{j} whenever i > j (if t_{i} = t_{j}, then y_{i} − y_{j} is an integer k in the interval [0, d]. This contradicts with the assumption that y_{1}, ⋯, y_{n} ∈ Δ′_{1}). Hence there is a unique permutation π such that

0 < t_{π(1)} < t_{π(2)} < ⋯ < t_{π(n)} < 1.

It is also clear that the connected component of Δ′_{1} determined by the point y_{1}, ⋯, y_{n} is the set of points y′_{1}, ⋯, y′_{n} such that

0 < y′_{π(1)} − [y′_{π(1)}] < ⋯ < y′_{π(n)} − [y′_{π(n)}] ≤ 1.

This is clearly the interior of a simplex with integral vertices and with determinant 1. The claim is proved.

For each fixed pair (i, j) with i > j, the hyperplanes H(i, j; k) (0 ≤ k ≤ d) form a sequence of successive parrallel hyperplances. Using the claim together with part (ii) of Theorem 7.4, by induction we deduce that Conjecture 1 is true for Δ. Theorem 7.5 is proved.

For applications to the zeta functions of projective hypersurfaces of degree d, we need to consider the case that Δ is the n-dimensional simplex in R^{n+1} generated by the origin and V + de_{i}. This Δ is a codimension 1 face of the polyhedron treated in Theorem 7.5. The boundary decomposition theorem (Corollary 5.2) shows that Conjecture 1 is also true for this Δ. Thus, we have

Corollary 7.6. Let Δ be generated by the origin and V + de_{i}, where d is a positive integer and V is a lattice point on the hyperplane x_{n+1} = D. Then Conjecture 1 is true.

As a final application, we prove

Corollary 7.7. Conjecture 1 is true if n ≤ 2.

Proof. If n = 1, it is trivial. Assume now that n = 2. By Theorem 5.5, we may assume that the origin is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin. Let D be the denominator of Δ. Since n = 2, one sees that Δ must be a simplex. By a suitable coordinate change, we may then assume that Δ_{1} is the line segment connecting V_{1} = (D, k) and V_{2} = (D, k + l), where l is a positive integer. It is easy to see that the intersection of Δ_{1} with any integral unit line segment is integral. The corollary follows from part (i) of Theorem 7.4. One can also use part (ii) of Theorem 7.4 to get the conclusion.

§8. The Function L(f, t)

Let 0 ≤ h ≤ n, and let f ∈ F_{q}[x_{1}, ⋯, x_{n}, (x_{1} ⋯ x_{h})^{−1}] be a Laurent polynomial. One can define the exponential sum

S_{m}(f) = ∑ Ψ(Tr_{F_{q^{m}} /F_{p}}(f(x_{1}, ⋯, x_{n}))),  (8.1)

where the sum is over all (x_{1}, ⋯, x_{n}) ∈ (F_{q^{m}})^{n} such that x_{j} ≠ 0 (1 ≤ j ≤ h). If h = n, S_{m}(f) reduces to S_{m}^{∗}(f). The associated L-function is

L(f, t) = exp( ∑_{m=1}^{∞} S_{m}(f)\frac{t^{m}}{m} ).  (8.2)

In this section, we explain how the results in §5 − 7 carry over to the function L(f, t)^{(−1)^{n−1}} and give several applications to zeta functions.

Let U be a subset of {1, 2, ⋯, n}, S_{1} = {1, 2, ⋯, h} and S_{2} = {h + 1, ⋯, n}. Let C(U) = {r ∈ C(Δ) | r_{i} > 0 if i ∈ U ∩ S_{2}}. Let A_{1}(U) denote the semilinear nuclear matrix (a_{s,r}(f)), where s and r run through the lattice points in the cone C(U). Let

A_{a}(U) = ∏_{i=0}^{a−1} A_{1}(U)^{τ^{i}} = A_{1}(U)A_{1}^{τ}(U) ⋯ A_{1}^{τ^{a−1}}(U).  (8.3)

The Dwork trace formula [2] asserts that

L(f, t)^{(−1)^{n−1}} = ∏_{l=0}^{n} ∏_{|U|=l} det^{τ^{−1}}(I − tq^{n−l}A_{a}(U))^{(−1)^{n−l}}.  (8.4)

Let

W_{U}(k) = card\{r ∈ C(U) ∩ Z^{n} | w(r) = \frac{k}{D(Δ)}\}.  (8.5)

Then the Newton polygons of det(I −tA_{a}(U)) computed with respect to q and the Newton polygon of det(I − tA_{1}(U)) computed with respect to p lie above the polygon with vertices (0, 0) and

(∑_{k=0}^{m} W_{U}(k), \frac{1}{D(Δ)} ∑_{k=0}^{m} kW_{U}(k)), m = 0, 1, 2, · · · .  (8.6)

Furthermore (Theorem 2.4), one of the above Newton polygons coincides with (8.6) if and only if the other Newton polygon coincides with (8.6). The degree polygon of det(I − tA_{1}(U)) with respect to a set T of lattice points V_{i} is defined in §4. Then all of the results in §5 − 7 carry over to det(I −tA_{1}(U)). For applications to L(f, t), one can then use (8.4). In particular, Theorem 3 is true for L(f, t). That is, there is a positive integer D^{∗} depending only on Δ such that if p ≡ 1 (mod D^{∗}) and p is large, then the generic Newton polygon of L(f, t)^{(−1)^{n−1}} coincides with the lower bound of Adolphson and Sperber (provided, of course, L(f, t)^{(−1)^{n−1}} is a polynomial for such general f. Adolphson-Sperber [2] showed that this is true if f is non-degenerate and commode with respect to S_{2}).

We now give several applications to zeta functions. First, let us consider affine hypersurfaces. Let d, d_{1}, ⋯, d_{n} be n positive integers. Let g(x_{1}, ⋯, x_{n}) be a general polynomial of totoal degree d or a general polynomial of degree d_{i} with respect to x_{i}. It is well-known [2] that the zeta function of the affine hypersurface defined by g(x) = 0 is given by

Z(g(x), qt) = L(A^{n+1}, x_{n+1}g(x); t) = \frac{L(A^{n} × G_{m}, x_{n+1}g(x); t)}{1 − q^{n}t}.

One checks that for all large prime number p, g(x) is non-degenerate and commode with respect to x_{1}, ⋯, x_{n} (in the terminology of [2]). Thus, the theorem of Adolphson and Sperber implies that for all large p, L(A^{n} × G_{m}, x_{n+1}g(x); t)^{(−1)^{n}} is a polynomial of a certain degree whose Newton polygon lies above the lower bound of Adolphson and Sperber. Now, Theorem 7.3 and Theorem 7.5 with D = 1 show that the Newton polygon of L(A^{n} × G_{m}, x_{n+1}g(x); t)^{(−1)^{n}} coincides with its lower bound for all large p. This proves Theorem 6 and Theorem 7.

Next, we consider projective hypersurfaces. Let g(x_{1}, ⋯, x_{n}) be a homogeneous polynomial of degree d and n variables. Let X be the hypersurface in P^{n−1} defined by g(x) = 0. Assume that X is smooth. Then, it is well-known that the zeta function for X has the form

Z(X, t) = \frac{R(t)^{(−1)^{n−1}}}{(1 − t)(1 − qt) ⋯ (1 − q^{n−2}t)},

where R(t) is a polynomial. The relation between R(t) and L-function [2] is known to be

L(A^{n} × G_{m}, x_{n+1}g(x); t)^{(−1)^{n}} = \frac{R(qt)}{R(q^{2}t)}.  (8.7)

We now derive an expression of R(t) in terms of Fredholm determinants. Note that Δ(x_{n+1}g) in R^{n+1} is n-dimensional, being contained in the hyperplane dx_{n+1} = x_{1} + ⋯ + x_{n}. In the notation of (8.4) with f = x_{n+1}g(x) (now, S_{2} = {1, 2, ⋯, n} and S_{1} = {n + 1}), we have

L(A^{n} × G_{m}, x_{n+1}g(x); t)^{(−1)^{n}} = ∏_{l=0}^{n+1} ∏_{|U|=l} det^{τ^{−1}}(I − tq^{n+1−l}A_{a}(U))^{(−1)^{n+1−l}},  (8.8)

where U runs over subsets of {1, 2, ⋯, n, (n + 1)}. Let

S(t) = ∏_{l=0}^{n} ∏_{|U_{1}|=l} det^{τ^{−1}}(I − tq^{n−l}A_{a}(U_{1}))^{(−1)^{n−l}},  (8.9)

where U_{1} runs only over subsets of {1, 2, ⋯, n} (corresponding to the variables x_{i} with 1 ≤ i ≤ n). If n + 1 ∈ U, then U ∩ S_{2} = {U − {n + 1}} ∩ S_{2}. Thus, C(U) = C(U − {n + 1}) and A_{1}(U) = A_{1}(U − {n + 1}). Then, (8.8) takes the form

L(A^{n} × G_{m}, x_{n+1}g(x); t)^{(−1)^{n}} = \frac{S(t)}{S(qt)}.  (8.10)

Let G(T) = S(t)/R(qt). Comparing (8.10) with (8.7), we deduce that G(t) = G(qt). Since G(t) is a non-zero p-adic meromorphic function with constant term 1. This forces G(t) = 1. That is, S(t) = R(qt) is a polynomial of expected degree. Theorem 7.6 with D = 1 and equation (8.9) then imply that the generic Newton polygon of S(t) and R(qt) coincides with its lower bound. This gives a p-adic proof of Theorem 5.

9. Newton Polygons and Frobenius Cycles on Lattice Points

In this section, we give a conjecture which describes all the large prime numbers p for which the Newton polygon coincides generically with its lower bound. This conjecture involves Frobenius cycles defined below. We will work on the chain level.

We assume that the origin 0 is a vertex of Δ and that Δ has only one face Δ_{1} of codimension 1 which does not contain the origin. Let V_{1}, ⋯, V_{J} be J fixed lattice points on the face Δ_{1} which include all vertices of Δ_{1}. Then Δ is generated by 0 and the V_{j}. Let E(V) be the set of all non-negative integral linear combinations of the V_{i}. One sees that E(V) is the monoid generated by the V_{i}. It is not a free monoid unless Δ is a simplex and the V_{i} are vertices. Let E(Δ) be the lattice points in the cone C(Δ). Then, E(V) is a submonoid of full rank in E(Δ). Let E_{≤k}(Δ) be the subset of E(Δ) consisting of all lattice points with weight at most k/D.

Let (u_{1}, ⋯, u_{k}) be an ordered set of k distinct lattice points in E(Δ). If

pu_{1} − u_{2} ∈ E(V), pu_{2} − u_{3} ∈ E(V), ⋯, pu_{k} − u_{1} ∈ E(V),

we call (u_{1}, ⋯, u_{k}) a Frobenius k-cycle. In particular, a Frobenius 1-cycle is a lattice point u_{1} such that (p − 1)u_{1} ∈ E(V). A set of lattice points is said to have a disjoint Frobenius covering if the set can be written as a disjoint union of Frobenius cycles.

Let f be the general Laurent polynomial f = ∑_{j} a_{j}x^{V_{j}}. Clearly, the Newton polyhedron of f is Δ. Recall that A_{1}(f) is the semi-linear nuclear Frobenius matrix associated with f.

Conjecture 9.1. Let m be a positive integer and p be a large prime number depending only on Δ. Then the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound at the m^{th} vertex if and only if the set E_{≤m}(Δ) has a disjoint covering by Frobenius cycles.

It can be proved that the conjecture is true if the V_{i} are vertices of a simplex. For the general case, we have the following partial result which asserts that the condition is necessary. The proof is based on the trivial observation that in order for a determinant to be non-zero, at least one term of the determinant expansion must be non-zero.

Theorem 9.2. If the generic Newton polygon of det(I −tA_{1}(f)) coincides with its lower bound at the m^{th} vertex, then the set E_{≤m}(Δ) has a disjoint covering by Frobenius cycles.

Proof. Recall that

A_{1}(f) = (a_{s,r}(f)) = (F_{ps−r}(f)π^{(w(r)−w(s))}),

where F_{u}(f) is defined by (2.9). The Fredholm determinant of a nuclear matrix can be expanded in the usual way. Write

det(I − tA_{1}(f)) = ∑_{k=0}^{∞} c(k)t^{k}.

Then

c(k) = (−1)^{k} ∑ ∑_{σ} ϵ(σ) ∏_{i=1}^{k} a_{s_{i},σ(s_{i})}(f),

where the outer sum is over all sets of k distinct lattice points in E(Δ), the inner sum is over all permutations on k letters, and ϵ(σ) is the sign of the permutation.

The m^{th} vertex of the lower bound for the Newton polygon is

(N(m), M(m)) = ( ∑_{k≤m/D} W (k), \frac{1}{D} ∑_{k≤m/D} kW (k)).

The block form of A_{1}(f) shows that we can write

c(N(m)) = (−1)^{N(m)} ∑_{σ} ϵ(σ) ∏_{i=1}^{N(m)} a_{s_{i},σ(s_{i})}(f) + o(p^{M(m)}),  (9.1)

where s_{1}, ⋯, s_{N(m)} are the elements of the set E_{≤m}(Δ) and the ord_{p} of the second term is greater than M(m). By assumption, the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound at the m^{th} vertex. This means that ord_{p}c(N(m)) = M(m). Thus, at least one of the products in (9.1) is non-zero. Fix such a term corresponding to some permutation σ. Since σ decomposes into disjoint cycles, the non-zero term decomposes into a product of the form

a_{u_{1},u_{2}}(f) ⋯ a_{u_{k},u_{1}}(f), σ(u_{1}) = u_{2}, ⋯, σ(u_{k}) = u_{1}.

Since these products are non-zero,

F_{pu_{1}−u_{2}}(f) ≠ 0, ⋯, F_{pu_{k}−u_{1}}(f) ≠ 0.

By the definition of F_{u}(f), this means that

pu_{1} − u_{2} ∈ E(V), pu_{2} − u_{3} ∈ E(V), ⋯, pu_{k} − u_{1} ∈ E(V).

Thus, E_{≤m}(V) has a disjoint Frobenius covering. The theorem is proved.

Example 9.3. The quotient monoid E(Δ)/E(V) is finite. Hence, it is a finite abelian group. Denote its annihilator by D^{∗}(Δ). If now p ≡ 1 (mod D^{∗}(Δ)), then (p−1)u ∈ E(V) for all u ∈ E(Δ). Thus, E_{≤m}(Δ) has a disjoint covering by Frobenius 1-cycles. Conjecture 9.1 would then imply that for all p ≡ 1 (mod D^{∗}(Δ)), the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound.

Example 9.4. Let f(x) be the general Laurent polynomial f(x) = ∑_{j=1}^{J} a_{j}x^{V_{j}}. For positive real number x, let π_{V} (x) be the number of prime numbers p ≤ x such that the generic Newton polygon of det(I − tA_{1}(f)) coincides with its lower bound. Then Conjecture 9.1 would imply that there is a positive rational number r such that

π_{V} (x) = r \frac{x}{log x} + O(\frac{x}{log^{2} x}).

This means for general f(x), the set of ordinary primes has positive rational density.

UNIVERSITY OF NEVADA, LAS VEGAS, NEVADA 89154 E-mail: dwan@nevada.edu

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