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1. On Hilbert's irreducibility theorem, Journal of No. Theory 6 (1974), 211–232. Written in 1969 -- but held up in publication by the inept editorship of Zassenhaus. It represents a transition from the topics of my UM thesis, to more sophisticated use of Riemann's Existence Theorem in solving Davenport's and Schinzel's problems and in introducing Galois stratification from my solving an Ax-Kochen problem. Developing Moduli spaces, to use Hurwitz spaces to generalize modular curve thinking had only started.
  1. Hilbert's Irreducibility Theorem: Start of the Hilbert-Siegel problems
  2. Non-regular anaglog of the Cebotarev Density Theorem and value sets of polynomial over finite fields
  3. A proof of the irreducibility theorem: By which we reduce a version of HIT to Chebotarev
The major point of §2 is its direct dealing with non-absolutely irreducible covers. For example: In the Galois stratification application, going to the Galois closure significantly introduced extension of constants which others were always assuming didn't exist. HIT74.pdf

2. PrincIdealsSmoothCurves1982.pdf

3. with J. Smith, Irreducible discriminant components of coefficient spaces, Acta Arithmetica XLIV (1984), 59–72. The Noether cover of affine n-space by affine n-space in respective variables y=(y1,…,yn) and x=(x1,…,xn) is the map that sends the y to the symmetric functions in (y1,…,yn). The discriminant locus is the image in the range of the points in the domain in which two or more entries are equal. Let I be a subset of {1,…,n}. Define the coefficient locus X(I) to be the locus defined by the equations xi = 0 for all i in I. We identify the irreducible components of the intersection of X(I) and the discriminant locus.

Theorem 3.1: If the elements of I have no common divisor, excluding some trivial hyperplane components this intersection if irreducible. There is an explicit bound A(n) so this result holds for positive characteristic if the characteristic exceeds A(n). IrrDiscComps-CoeffSps1984.pdf

4. The (g - 1)-support cover of the Canonical Locus, with H. Farkas, Max Noether characterized non-hyperelliptic curves S of genus g by this property. The vector space Hqof holomorphic q-differentials, q an integer > 1, is spanned by q-fold products of holomorphic differentials chosen from a (fixed) basis of the holomorphic differentials on S. Noether's proof depended on showing that for a non-hyperelliptic surface there always exists a positive divisor D of degree g - 2 on S, such that for any s ∈ S there is exactly one holomorphic differential (up to multiplication by a constant) whose divisor contains D + s in its support.

An alternate proof of Noether's theorem based on the following result has been suggested: S is non-hyperelliptic if and only if S has a holomorphic differential θ with these properties.

  1. Its support consists of 2g -2 distinct points on S.
  2. No proper subset of the support appears as the support of any other holomorphic differential.

The existence of θ and Noether's theorem follows from this. Denote by S(g-1) the g-1 symmetric product of S. Consider
ψ: S(g-1)×S(g-1) → S(2g-2) by addition of divisors.

Main Theorem: Take Z the subset of the right side consisting of divisors of holomorphic differentials. Then, for g ≥ 3, ψ-1(Z) is irreducible if and only if S is not hyperelliptic. The proof characterizes when the branch loci of Sg-1 → S(g-1) given by both projections from S(g-1)×S(g-1) are the same. g-1SuppCoverJdAnal86.pdf

5. On the Sprindzuk-Weissauer approach to universal Hilbert subsets, Israel Journal of Mathematics 51 (1985), 347–363. The works of both Sprindzuk (1979-80) and Weissauer (1980) consider the relation between Hilbert subsets of Q and sets consisting of powers of primes. A comparison of their results leads to generalizations and new proofs devoid of either p-adic diophantine approximation or of nonstandard arithmetic (§ 3 and § 4). Results of Weissauer, giving new Hilbertian infinite extensions of every Hilbertian field, receive short direct standard proofs. It also gives a negative answer to a question of Roquette on the relation between Hilbert sets and value sets. I think this is the first paper to consider the idea of Universal Hilbert subsets.

One of main values of the paper is its use (and reminder of) Weil's distributions. That was the device by which he and C.L. Siegel proved two of their respective major results. Distributions put in arithmetic fashion the idea that a function on a compact Riemann surface is determined by its zeros and poles. Each of these have their own quasi-integral functions on rational points. Those functions then work in whatever other functions contain one of those zeros or poles. That is, you can use them to tear apart functions arithmetically. SprindWeiss85.pdf

6. with M. Jarden, σ-fields, PJM 185 (1998) 307–313. H(ilbert)'s I(rreducibility) T(heorem) is a diophantine limitation statement that applies to any number field. A field K satisfies HIT if for any finite collection of nonconstant maps (of degree at least 2) from algebraic curves, some values of K aren't in the image of any K points under these maps. The paper's main examples are g-Hilbertian fields. Such a field K is not the union of maps (at least degree 2) of K points from curves of genus at most g, yet there are g' > g for which this statement fails. sigmahilb.html %-%-% sigmahilb.pdf

7. The Domain for Hilbert's Irreducibility Theorem, This file leads to a discussion of many aspects of Hilbert's Theorem, the most general arithmetic result on specializing algebraic objects to particular cases. Classical Domain Subtopics: Its application to the Inverse Galois Problem, producing high rank elliptic curves over Q; using Siegel's Thm. for precise HIT results; Universal Hilbert Subsets; Serre's Open Image Theorem as an HIT archetype. General Diophantine domain Subtopics: Weissauer's Thm. and relation between HIT and other arithmetic field properties; and the use of Pseudo-Algebraically-Closed fields to investigate the absolute Galois group of Q. HIT-domain.html

8. with S. R. Valluri (West. Ont. Univ.), Chebychev Derived Spindown Parameters for Gravitational Wave Signals from Pulsars, Canadian J. Physics Vol. 86 2008. 597–600. As of 10/10/07, noone has yet detected gravity waves, though General Relativity has predicted them for 80 years. Their most likely identifiable source is spinning neutron stars – weighty, yet tinier than the earth, stellar objects – by integrating space-perturbating data over long periods (several months at least). Such integration must cancel inter-solar-system data, and account for the neutron spin slowing with time. Neutron stars are a final stage for solar masses between standard red-giants and black holes. Inserting into the master equations parameters to detect what is involved in their spin-down is the topic of this paper. April3CJPFriedValluri.pdf