A Chevalley-Warning Approach To P-adic Estimates Of Character Sums

Daqing Wan

Abstract. The elemetary Chevalley-Warning congruence method is applied to obtain several p-adic estimates of character sums over finite fields.

1. Introduction

Let F_{q} be the finite field of q elements with characteristic p. Let F_{i}(x_{1}, ⋯, x_{n}) (i = 1, ⋯, r) be polynomials of degree d_{i} over F_{q}. The classical Chevalley-Warning theorem asserts that if n > ∑_{i} d_{i}, then the characteristic p divides the number N_{q}(F_{1}, ⋯, F_{r}) of F_{q}-rational solutions of the system

F_{1}(x_{1}, ⋯, x_{n}) = F_{2}(x_{1}, ⋯, x_{n}) = ⋯ = F_{r}(x_{1}, ⋯, x_{n}) = 0.  (1.1)

The proof of this theorem is very simple and elegant, see [8]. It is based on the following congruence formula

N_{q}(F_{1}, ⋯, F_{r}) ≡ ∑_{x∈F_{q}^{n}}(1 − F_{1}(x)^{q−1}) ⋯ (1 − F_{r}(x)^{q−1}) (mod p),  (1.2)

a principal first noted by Lebesgue in 1837 (see the notes of Chapter 7 in [5]).

The Chevalley-Warning theorem was greatly improved by Ax [3] (for the case r = 1) and Katz [4] (for general r). The Ax-Katz theorem asserts that if b is the least integer such that

b ≥ \frac{n − ∑_{i} d_{i}}{max_{i} d_{i}} ,  (1.3)

then q^{b} divides N_{q}(F_{1}, ⋯, F_{r}). Ax [3] also obtained a weaker result for general r, which replaces the right side of (1.3) by (n − ∑_{i} d_{i})/ ∑_{i} d_{i}. The Ax-Katz theorem is best possible

1991 Mathematics Subject Classification. Primary 11A07, 11D72, 11T23 in the sense that for each n and each multiple degree (d_{1}, ⋯, d_{r}), there are polynomials F_{i} of degree d_{i} such that the highest power of q dividing N_{q}(F_{1}, ⋯, F_{r}) is exactly q^{b}. Instead of using a congruence formula similar to (1.2), Ax used the well-known expression of the number N_{q}(F_{1}, ⋯, F_{r}) in terms of exponential sums. Ax’s proof is p-adic in nature and uses the Stickelberger theorem on Gauss sums. Even though there is a quick proof of the Chevalley-Warning theorem, Ax [3] said “there does not seem to be any simple proof of the fact that q divides N_{q}(F) if n > d”.

Katz’s proof (for general r) is much deeper and uses Dwork’s theory of completely continuous operators in a infinitely dimensional p-adic Banach space. The Ax-Katz theorem was generalized to exponential sums by Sperber [9] and Adolphson-Sperber [1-2]. Their proof is similar to Katz’s proof and uses Dwork’s p-adic theory. In [10], it was shown that the Katz theorem can be proved by using Ax’s method. Adolphson-Sperber [2] then realized that their theorem on exponential sums can also be proved by using Ax’s method.

Recently, motivated by coding theoretic considerations, Moreno-Moreno [6-7] have obtained a new theorem which improves the Ax-Katz theorem in certain special cases. Their idea is to reduce the question under consideration from F_{q} to the prime field F_{p}. Also motivated by coding theoretic considerations, Ward [11] recently found a new proof of the Ax theorem (the case r = 1). In comparison to Ax’s proof, Ward’s proof is closer in line to Chevalley-Warning’s proof. Essentially, Ward startes with the case r = 1 of the limiting congruence formula

N_{q}(F_{1}, ⋯, F_{r}) = ∑_{x∈T_{q}^{n}}(1 − ( lim_{k→∞} F_{1}(x)^{q^{k}})^{q−1}) ⋯ (1 − ( lim_{k→∞} F_{r}(x)^{q^{k}})^{q−1}),

where T_{q} is the Teichmüller lifting of F_{q} and F (x) has p-adic coefficients. Note that lim_{k→∞} a^{q^{k}} is simply the Teichmüller lifting of a ∈ F_{q}. Ward’s proof uses p-adic lifting and avoids the Stickelberger theorem. However, his proof depends on his polarization theory and is not very simple.

Motivated by Ward’s proof, in §2 we give a direct and simple proof of the Ax-Katz theorem in the prime field case. Combining with Moreno-Moreno’s reduction, this gives a simple proof of the Moreno-Moreno theorem and answers Ax’s questions. Our proof is parrallel to the Chevalley-Warning proof and uses only congruence formulas over the rational integers and monomial coefficients. The Stickelberger theorem, p-adic liftings and polarizations are not involved.

For a general finite field, the same congruence proof works if we replace the rational integers by algebraic integers in number fields or by p-adic integers in p-adic fields. Using the congruence argument, in §3 we generalize the Ax-Katz theorem to multiplicative character sums. Finally, in §4 we extend the result in §3 to mixed character sums, which includes Moreno-Moreno’s theorem on exponential sums as a special case.

Acknowledgment: I would like to thank Hendrik Lenstra for calling my attention to Moreno-Moreno’s work [6-7] and Ward’s work [11], which led to the present paper, and for pointing out that the Moreno-Moreno theorem does not imply the Ax-Katz theorem. Thanks are also due to the referee for several helpful comments.

2. The Ax-Katz Theorem In Prime Field Case

In this section, we give a direct and simple proof of the Ax-Katz theorem in the prime field case. Let F_{i}(x_{1}, ⋯, x_{n}) (1 ≤ i ≤ r) be polynomials of degree d_{i} with integral coefficients. We are interested in the number N_{p}(F_{1}, ⋯, F_{r}) of solutions of the congruence system

F_{1}(x_{1}, ⋯, x_{n}) ≡ F_{2}(x_{1}, ⋯, x_{n}) ≡ ⋯ ≡ F_{r}(x_{1}, ⋯, x_{n}) ≡ 0 (mod p).  (2.1)

Let b be the least integer satisfying (1.3), we need to prove that p^{b} divides N_{p}(F_{1}, ⋯, F_{r}).

Let S be the set consisting of zero and the g^{ip^{n}} (0 ≤ i ≤ p − 2), where g is a fixed primtive root modulo p^{n} if p > 2 and g = 1 if p = 2. Then S is a complete residue system modulo p. Similar to (1.2), we have the following well-known congruence formula

N_{p}(F_{1}, ⋯, F_{r}) ≡ ∑_{x∈S^{n}}(1 − F_{1}(x)^{(p−1)p^{n}}) ⋯ (1 − F_{r}(x)^{(p−1)p^{n}}) (mod p^{n}).  (2.2)

Expanding (2.2) and by induction on r, we see that it suffices to prove that

A = ∑_{x∈S^{n}} ∏_{i=1}^{r} F_{i}(x)^{(p−1)p^{n}} ≡ 0 (mod p^{b}).  (2.3)

Let F_{i}(x) = ∑_{j=1}^{m_{i}} a_{ij}x^{e_{ij}}, where the e_{ij} are vectors in Z_{≥0}^{n} whose sums of coordinates are at most d_{i}. Expanding (2.3) and interchange the summation, we have

A = ∑_{k_{i1}+···+k_{im_{i}} =(p−1)p^{n},1≤i≤r} ∏_{i=1}^{r} \binom{(p − 1)p^{n}}{k_{i1}, ⋯, k_{im_{i}}}(∏_{i=1}^{r} ∏_{j=1}^{m_{i}} a_{ij}^{k_{ij}}) ∑_{x∈S^{n}} x^{∑_{i,j} k_{ij}e_{ij}}. (2.4)

By the classical formula of Legendre, ord_{p}(k!) = (k −σ(k))/(p−1), where σ(k) denotes the sum of the digits in the base p expansion of k. It follows that the p-order of the monomial coefficient in (2.4) is

\frac{1}{p − 1} ∑_{i=1}^{r}((p − 1)p^{n} − (p − 1) − ∑_{j=1}^{m_{i}}(k_{ij} − σ(k_{ij}))) = \frac{1}{p − 1} ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − (p − 1)).

By our choice of the complete residue system S, the following is valid:

∑_{t∈S} t^{k} ≡ \{ \begin{matrix}(g^{k(p−1)p^{n}} − 1)/(g^{kp^{n}} − 1) ≡ 0 (mod p^{n}), & if (p − 1) does not divide k \\ p (mod p^{n}), & if k = 0 \\ ∑_{i=0}^{p−2} g^{kip^{n}} ≡ p − 1 (mod p^{n}), & if (p − 1) divide k and k > 0. \end{matrix}

(2.5)

Thus, in (2.4) we need only to check those terms for which

∑_{i,j} k_{ij}e_{ij} ≡ 0 (mod (p − 1)),  (2.6)

where the congruence means that each coordinate of the vector is divisible by (p − 1). Assume that s of the coordinates in (2.6) are not numerically zero. By (2.4) and (2.5), we are reduced to proving that (noting that the number N_{p}(F_{1}, ⋯, F_{r}) is an integer)

\frac{n − ∑_{i} d_{i}}{max_{i} d_{i}} ≤ \frac{1}{p − 1} ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − (p − 1)) + (n − s).  (2.7)

Since k ≡ σ(k) (mod (p − 1)), we can replace k_{ij} by σ(k_{ij}) in (2.6):

∑_{i,j} σ(k_{ij})e_{ij} ≡ 0 (mod (p − 1)).  (2.8)

Furthermore, s of the coordinates in (2.8) are not numerically zero. Adding these coordinates and letting d = max_{i} d_{i}, we deduce that

s(p − 1) − (p − 1) ∑_{i=1}^{r} d_{i} ≤ ∑_{i=1}^{r} d_{i}(∑_{j=1}^{m_{i}} σ(k_{ij}) − (p − 1))
≤ d ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − (p − 1)).  (2.9)

This inequality implies that

\begin{aligned}\frac{1}{p − 1} ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − (p − 1)) + (n − s) & ≥ (\frac{s − ∑_{i} d_{i}}{d}) + (n − s) \\ & ≥ (\frac{n − ∑_{i} d_{i}}{d}). \end{aligned}

By (2.7), the Ax-Katz theorem is proved for the prime field F_{p}.

3. Multiplicative Character Sums

In this section, we generalize the Ax-Katz theorem to multiplicative character sums. Let K_{q} be the unique unramified extension of degree f over the p-adic rational number field Q_{p}. Let T_{q} be the set consisting of the roots of t^{q} = t in K_{q}. Then, the reduction of T_{q} modulo p is the finite field F_{q}. Let T be the Teichmüller character, i.e., T (t̄) = t. This is a primitive multiplicative character on F_{q} of order (q − 1). Any multiplicative character of F_{q} is a power of T. Let χ_{i}(t) = T (t)^{j_{i}} (1 ≤ i ≤ r) be multiplicative characters of F_{q}, where the j_{i} are integers satisfying 0 ≤ j_{i} ≤ q − 1. By convention, T^{0}(a) is the character with value 1 for all a ∈ F_{q}; while T^{q−1}(a) is the character with value 1 for all a ∈ F_{q}^{∗} and T^{q−1}(0) = 0. Let F_{i}(x_{1}, ⋯, x_{n}) be polynomials of degree d_{i} over F_{q}. Define a character sum by

S_{q}(χ, F) = ∑_{x∈F_{q}^{n}} χ_{1}(F_{1}(x)) ⋯ χ_{r}(F_{r}(x)).  (3.1)

For an integer k ≥ 0, define σ_{q}(k) to be the sum of the digits in the expansion of k as a base q number. If q = p, then σ_{q}(k) = σ(k). For a real number x, define (x)^{∗} to be the smallest integer not less than x. We have the following theorem.

Theorem 3.1. Let d = max_{i} d_{i} and q = p^{f}. Then, the q-order of S_{q}(χ, F) is at least

\frac{1}{f} ∑_{a=0}^{f−1}(\frac{n − \frac{1}{(q−1)} ∑_{i=1}^{r} σ_{q}(p^{a}j_{i})d_{i}}{d})^{∗}.  (3.2)

Proof. To simpliyfy notations, suppose that the F_{i}(x) are already lifted to K_{q}. Since T (x̄) ≡ x^{q^{n}} (mod q^{n}) for all integral x ∈ K_{q}, similar to (1.2), we have the following congruence formula

S_{q}(χ, F) ≡ ∑_{x∈T_{q}^{n}} F_{1}(x)^{j_{1}q^{n}} ⋯ F_{r}(x)^{j_{r}q^{n}} (mod q^{n}).  (3.3)

Let F_{i}(x) = ∑_{j=1}^{m_{i}} a_{ij}x^{e_{ij}}, where the e_{ij} are vectors in Z_{≥0}^{n} whose sums of coordinates are at most d_{i} and the a_{ij} are p-adic integers in K_{q}. Expanding (3.3) and interchange the summation, we have the following congruence modulo q^{n}:

S_{q}(χ, F) ≡ ∑_{k_{i1}+···+k_{im_{i}} =j_{i}q^{n},1≤i≤r} ∏_{i=1}^{r} \binom{j_{i}q^{n}}{k_{i1}, ⋯, k_{im_{i}}}(∏_{i=1}^{r} ∏_{j=1}^{m_{i}} a_{ij}^{k_{ij}}) ∑_{x∈T_{q}^{n}} x^{∑_{i,j} k_{ij}e_{ij}}. (3.4)

By the classical formula of Legendre, ord_{p}(k!) = (k − σ(k))/(p − 1). It follows that the q-order of the monomial coefficient in (3.4) is

\frac{1}{f(p − 1)} ∑_{i=1}^{r}(j_{i}q^{n} − σ(j_{i}) − ∑_{j=1}^{m_{i}}(k_{ij} − σ(k_{ij}))) = \frac{1}{f(p − 1)} ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − σ(j_{i})). (3.5)

By the definition of T_{q}, the following is valid:

∑_{t∈T_{q}} t^{k} = \{ \begin{matrix}0, & if (q − 1) does not divide k \\ q, & if k = 0 \\ q − 1, & if (q − 1) divides k and k > 0. \end{matrix}  (3.6)

Thus, in (3.4) we need only to check those terms for which

∑_{i,j} k_{ij}e_{ij} ≡ 0 (mod (q − 1)),  (3.7)

where the congruence means that each coordinate of the vector is divisible by (p−1). Since k ≡ σ_{q}(k) (mod (q − 1)), by (3.7) we have

∑_{i,j} σ_{q}(k_{ij})e_{ij} ≡ 0 (mod (q − 1)).  (3.8)

Assume that s of the coordinates in (3.7) are not numerically zero. The definition of σ_{q}(k) shows that s of the coordinates in (3.8) are not numerically zero. Adding these coordinates and letting d = max_{i} d_{i}, we deduce that

s(q − 1) − ∑_{i=1}^{r} j_{i}d_{i} ≤ ∑_{i=1}^{r} d_{i}(∑_{j=1}^{m_{i}} σ_{q}(k_{ij}) − j_{i})
≤ d ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ_{q}(k_{ij}) − j_{i}).  (3.9)

Since ∑_{j} k_{ij} = j_{i}q^{n} ≥ 0 and σ_{q}(j_{i}) = j_{i} for all i, we deduce that for all i,

∑_{j} σ_{q}(k_{ij}) − j_{i} ≡ 0 (mod (q − 1)).  (3.10)

It then follows from (3.9) and (3.10) that

(\frac{s − \frac{1}{q−1} ∑_{i} j_{i}d_{i}}{d})^{∗}(q − 1) ≤ ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ_{q}(k_{ij}) − j_{i}).  (3.11)

Let a be a non-negative integer. If we multiply both sides of (3.7) by p^{a}, then (3.8) and (3.10) remain true with σ_{q}(k_{ij}) replaced by σ_{q}(p^{a}k_{ij}) (and j_{i} replaced by σ_{q}(p^{a}j_{i})). Furthermore, s of their coordinates are not numerically zero. Thus, similar to (3.11), we have

(\frac{s − \frac{1}{q−1} ∑_{i} σ_{q}(p^{a}j_{i})d_{i}}{d})^{∗}(q − 1) ≤ ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ_{q}(p^{a}k_{ij}) − σ_{q}(p^{a}j_{i})).  (2.12)

Adding equation (2.12) for a = 0, 1, ⋯, f − 1, we deduce that

∑_{a=0}^{f−1}(\frac{s − \frac{1}{q−1} ∑_{i} σ_{q}(p^{a}j_{i})d_{i}}{d})^{∗}(q − 1) ≤ \frac{q − 1}{p − 1} ∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − σ(j_{i})),  (3.13)

where we used the simple fact that for all integers k ≥ 0,

∑_{a=0}^{f−1} σ_{q}(p^{a}k) = σ(k)\frac{q − 1}{p − 1}.

By (3.5) and (3.6), we conclude that the q-order of each term in (3.4) is at least

min_{0≤s≤n}\{\frac{1}{f} ∑_{a=0}^{f−1}(\frac{s − \frac{1}{q−1} ∑_{i} σ_{q}(p^{a}j_{i})d_{i}}{d})^{∗} + (n − s)\} = \frac{1}{f} ∑_{a=0}^{f−1}(\frac{n − \frac{1}{q−1} ∑_{i} σ_{q}(p^{a}j_{i})d_{i}}{d})^{∗}.

The theorem is proved.

If we take the χ_{i} to be the trivial characters with j_{i} = q − 1 for all i, then the identity p^{a}(q − 1) = (p^{a} − 1)q + (q − p^{a}) shows that σ_{q}(p^{a}j_{i}) = q − 1 for all 0 ≤ a ≤ q − 1. In this case, the number in (3.2) is reduced to the integer b in (1.3) and

N_{q}(F_{1}, ⋯, F_{r}) = ∑_{x∈F_{q}^{n}}(1 − χ_{1}(F_{1}(x))) ⋯ (1 − χ_{r}(F_{r}(x))).

Thus, Theorem 3.1 includes the Ax-Katz theorem as a special case. Using the inequality (x)^{∗} + (y)^{∗} ≥ (x + y)^{∗} and the identity ∑_{a=0}^{f−1} σ_{q}(p^{a}j) = σ(j)(q − 1)/(p − 1), we obtain

Corollary 3.2. The q-order of S_{q}(χ, F) is at least

\frac{1}{f} (\frac{nf − \frac{1}{(p−1)} ∑_{i} σ(j_{i})d_{i}}{d})^{∗}.  (3.14)

4. Mixed Character Sums

In this section, we prove that similar results are true for mixed character sums. We shall combine the congruence method, Moreno-Moreno’s reduction and Ax’s method. An alternative approach (not discussed here) would be to replace the congruence method by Jacobi sums and Stickelberger’s theorem. Let χ_{i} = T^{j_{i}} (1 ≤ i ≤ r) be multiplicative characters of F_{q} as above. Let ψ be a fixed non-trivial additive character of F_{q}. Let F_{i}(x_{1}, ⋯, x_{n}) (1 ≤ i ≤ r + 1) be polynomials of degree d_{i} over F_{q}. Define a mixed character sum by

S_{q}(χ, ψ, F) = ∑_{x∈F_{q}^{n}} χ_{1}(F_{1}(x)) ⋯ χ_{r}(F_{r}(x))ψ(F_{r+1}(X)).  (4.1)

For 1 ≤ i ≤ r + 1, let

h_{i} = max_{(k_{1},···,k_{n})} σ(k_{1}) + ⋯ + σ(k_{n}),

where the maximun is taken over the degrees of all monomials x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}} in F_{i}. We have the following theorem.

Theorem 4.1. The p-order of S_{q}(χ, ψ, F) is at least

\frac{f}{max_{1≤i≤r+1} h_{i}} (n − \frac{1}{f(p − 1)} ∑_{i=1}^{r} σ(j_{i})h_{i}).  (4.2)

Proof. We assume that the F_{i}(x) are already lifted to a polynomial in K_{q}[x] of degree d_{i}. Similar to §3, we have the following congruence formula

S_{q}(χ, ψ, F) ≡ ∑_{x∈T_{q}^{n}} F_{1}(x)^{j_{1}q^{n}} ⋯ F_{r}(x)^{j_{r}q^{n}}ψ(F_{r+1}(x)) (mod q^{n}),  (4.3)

where for simplicity, ψ(F_{r+1}(x)) means the value of ψ at the reduction of F_{r+1}(x) modulo p. We use Moreno-Moreno’s reduction to reduce the above sum to a sum over T_{p}^{n}. Choose elements α_{1}, ⋯, α_{f} in T_{q} such that their reduction is a basis of F_{q} over F_{p}. Then, every element x_{i} of T_{q} can be uniquely written in the form x_{i} ≡ y_{i1}α_{1} + ⋯ + y_{if}α_{f}(mod p), where the y_{ij} are elements in T_{p}. Let k = k_{0} + k_{1}p + k_{2}p^{2} + ⋯ be a positive integer. Then,

x_{i}^{k} ≡ (∑_{j=1}^{f} y_{ij}α_{i})^{k_{0}+k_{1}p+k_{2}p^{2}+···}
≡ (∑_{j=1}^{f} y_{ij}α_{i})^{k_{0}}(∑_{j=1}^{f} y_{ij}α_{i}^{p})^{k_{1}} ⋯ (mod p).  (4.4)

Thus, we can replace the polynomial F_{r+1}(x) of degree d_{r+1} by a p-adic integral polynomial G_{r+1}(y) in K_{q}[y_{11}, y_{12}, ⋯, y_{nf}] of degree at most h_{r+1}, and the variables y_{ij} take values in T_{p}. Since the y_{ij} are in T_{p}, we have ψ(G_{r+1}(y)) = ψ_{p}(tr(G_{r+1}(y))) = ψ_{p}(G′_{r+1}(y)), where ψ_{p} is an additive character ψ_{p} of F_{p} and the polynomial G′_{r+1}(y) has coefficients in Z_{p}. For each 1 ≤ i ≤ r, let j_{i} = j_{i}(0) + j_{i}(1)p + ⋯ + j_{i}(f − 1)p^{f−1} be the base p expansion of j_{i}. Then, the congruence reduction idea as in (4.4) shows that we can replace each polynomial F_{i}^{j_{i}}(x) (coming from χ_{i}(F_{i}) = T (F_{i}^{j_{i}})) by a product polynomial ∏_{k=0}^{f−1} G_{ik}^{j_{i}(k)}(y), where each G_{ik} is a p-adic integral polynomial in K_{q}[y_{11}, y_{12}, ⋯, y_{nf}] of degree at most h_{i}, and the variables y_{ij} take values in T_{p}. Thus, we are reduced to the case f = 1 except that the polynomials F_{i}(x) (1 ≤ i ≤ r) may have coefficients in the extension K_{q}. Namely, we are reduced to consider

S_{p}(χ, ψ, F) ≡ ∑_{x∈T_{p}^{n}} F_{1}(x)^{j_{1}p^{n}} ⋯ F_{r}(x)^{j_{r}p^{n}}ψ_{p}(F_{r+1}(x)) (mod p^{n}),  (4.5)

where 0 ≤ j_{i} ≤ p − 1, each polynomials F_{i}(x) has at most degree d_{i} with p-adic integral coefficients in the extension K_{q} and F_{r+1}(x) has coefficients in Z_{p}. We need to prove that the p-order of the sum in (4.5) is at least

\frac{1}{max_{1≤i≤r+1} d_{i}} (n − \frac{1}{(p − 1)} ∑_{i} j_{i}d_{i}).  (4.6)

For 1 ≤ i ≤ r, let F_{i}(x) = ∑_{j=1}^{m_{i}} a_{ij}x^{e_{ij}}, where the e_{ij} are vectors in Z_{≥0}^{n} whose sums of coordinates are at most d_{i}. Let F_{r+1}(x) = ∑_{j=1}^{m} b_{j}x^{e_{j}}, where the e_{j} are vectors in Z_{≥0}^{n} whose sums of coordinates are at most d_{r+1}. The multiplicative part can be expanded as before:

∏_{i=1}^{r} F_{i}(x)^{j_{i}p^{n}} = ∑_{k_{i1}+···+k_{im_{i}} =j_{i}p^{n},1≤i≤r} ∏_{i=1}^{r} \binom{j_{i}p^{n}}{k_{i1}, ⋯, k_{im_{i}}}(∏_{i=1}^{r} ∏_{j=1}^{m_{i}} a_{ij}^{k_{ij}})x^{∑_{i,j} k_{ij}e_{ij}}. (4.7)

The p-order of the monomial coefficient in (4.7) is computed to be

\frac{1}{p − 1} ∑_{i=1}^{r}(j_{i}p^{n} − j_{i} − ∑_{j=1}^{m_{i}}(k_{ij} − σ(k_{ij}))) = \frac{1}{p − 1} ∑_{i}(∑_{j} σ(k_{ij}) − j_{i}).  (4.8)

For the additive part, we use Gauss sums and Stickelberger’s theorem. For integer k with 0 ≤ k ≤ p − 2, define the Gauss sum

g_{k} = ∑_{x∈T_{p}} t^{p−1−k}ψ_{p}(x).

Let ∑_{k} G(k)t^{k} be a polynomial of degree p − 1 such that

ψ_{p}(t̄) = ∑_{k=0}^{p−1} G(k)t^{k},  (4.9)

for all t ∈ T_{p}. One checks that G(0) = 1, G(p − 1) = −p/(p − 1) and for 1 ≤ k ≤ p − 2, G(k) = g_{k}/(p − 1). The Stickelberger theorem asserts that the p-order of g_{k} is k/(p−1) (this prime field case can be proved easily). Thus, the p-order of G(k) is k/(p−1) for all 0 ≤ k ≤ p − 1. For simplicity, we define ψ_{p}(x) = ψ_{p}(x̄) if x is a p-adic integer in Z_{p}. Using (4.9), for all x ∈ T_{p}^{n} we have the expansion

ψ_{p}(F_{r+1}(x)) = ∏_{j=1}^{m} ψ_{p}(b_{j}x^{e_{j}})
= ∑_{l_{1},···,l_{m}=0}^{p−1}(∏_{j=1}^{m} G(l_{j})) ∏_{j=1}^{m}(b_{j}x^{e_{j}})^{l_{j}}.  (4.10)

The p-order of the coefficient in (4.10) is (∑_{j=1}^{m} l_{j})/(p − 1).

Substituing (4.7) and (4.10) into (4.5), multiplying them out and using the definition of T_{q}, we need only to check those terms for which

∑_{i=1}^{r} ∑_{j=1}^{m_{i}} k_{ij}e_{ij} + ∑_{j=1}^{m} l_{j}e_{j} ≡ 0 (mod (p − 1)),  (4.11)

where the congruence means that each coordinate of the vector is divisible by (p − 1). Assume that s of the coordinates in (4.11) are not numerically zero and let d = max_{1≤i≤r+1} d_{i}. By the above computation, we are reduced to proving that

\frac{1}{d}(n − \frac{1}{(p − 1)} ∑_{i=1}^{r} j_{i}d_{i}) ≤ \frac{1}{p − 1}(∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − j_{i}) + ∑_{j=1}^{m} l_{j}) + (n − s).  (4.12)

Replacing k_{ij} by σ(k_{ij}) in (4.11), we get

∑_{i=1}^{r} ∑_{j=1}^{m_{i}} σ(k_{ij})e_{ij} + ∑_{j=1}^{m} l_{j}e_{j} ≡ 0 (mod (p − 1)).  (4.13)

Furthermore, s of the coordinates in (4.13) are not numerically zero. Adding these coordinates, we deduce that

s(p − 1) − ∑_{i=1}^{r} j_{i}d_{i} ≤ ∑_{i=1}^{r} d_{i}(∑_{j=1}^{m_{i}} σ(k_{ij}) − j_{i}) + ∑_{j=1}^{m} l_{j}d_{r+1}
≤ d(∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − j_{i}) + ∑_{j=1}^{m} l_{j}).  (4.14)

By (4.12), the p-order of S_{p}(χ, ψ, F) is at least

\frac{1}{p − 1}(∑_{i=1}^{r}(∑_{j=1}^{m_{i}} σ(k_{ij}) − j_{i}) + ∑_{j=1}^{m} l_{j}) + (n − s)

≥ (\frac{s − \frac{1}{p−1} ∑_{i=1}^{r} j_{i}d_{i}}{d}) + (n − s)

≥ \frac{1}{d}(n − \frac{1}{p − 1} ∑_{i=1}^{r} j_{i}d_{i}).

Theorem 4.1 is proved.

If we take the χ_{i} to be the characters with j_{i} = 0 and take F_{i} with d_{i} = 1 for all i ≤ r, then

S_{q}(χ, ψ, F) = S_{q}(ψ, F_{r+1}) = ∑_{x∈F_{q}^{n}} ψ(F_{r+1}(x))  (4.15)

is the exponential sum treated by Sperber [9]. Theorem 4.1 shows that the p-order of the exponential sum in (4.15) is at least fn/h_{r+1}. This is a theorem of Moreno-Moreno [6] on exponential sums, which improves a theorem of Sperber [9].

Corollary 4.2. The q-order of S_{q}(χ, ψ, F) is at least

\frac{1}{max_{1≤i≤r+1} d_{i}} (n − \frac{1}{f(p − 1)} ∑_{i=1}^{r} σ(j_{i})d_{i}).  (4.16)

References

[1] A. Adolphson and S. Sperber, p-Adic estimates for exponential sums and the theorem of Chevalley-Warning, Ann. Scient. E. N. Superior 4th. series, 20(1987), 545-556.

[2] A. Adolphson and S. Sperber, p-Adic estimates for exponential sums, Lecture Notes in Math., 1454. (1990), 11-22.

[3] J. Ax, Zeros of polynomials over finite fields, Amer. J. Math., 86(1964), 255-261.

[4] N.M. Katz, On a theorem of Ax, Amer. J. Math., 93(1971), 485-499.

[5] R. Lidl and H. Niederreiter, Finite Fields, Encycl. of Math and Its Appl. Addison-Wesley, 20, (1983).

[6] O. Moreno and C.J. Moreno, Improvements of the Chevalley-Warning and the Ax-Katz theorem, Amer. J. Math., to appear.

[7] O. Moreno and C.J. Moreno, The MacWilliams-Sloane conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes, submitted to IEEE Trans. Inform. Theory.

[8] W.M. Schmidt, Equations over finite fields, Lecture Notes in Math., 536, (1976), p.135.

[9] S. Sperber, On the p-adic theory of exponential sums, Amer. J. Math., 108(1986), 255-296

[10] D. Wan, An elementary proof of a theorem of Katz, Amer. J. Math., 111(1989), 1-8.

[11] H.N. Ward, Weight polarization and divisibility, Discrete Math., 83(1990), 315-326.

Department of Mathematical Sciences University of Nevada, Las Vegas Las Vegas, Nevada 89154 E-mail: dwan@nevada.edu