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- Leonid Vaserstein, Ethel R. Wheland

If a sum of univariate polynomials is zero, then there are restrictions on the multiplicities of zeros of the polynomials.

This paper was motivated by the following open problem ([8], p.390): “CNTA 5.15 (Frits Beukers). Prove or disprove the following statement: There exist four polynomials A,B,C,D with integer coefficients (in any number of variables) such that AD−BC = 1 and all integer solutions of ad−bc = 1 can be obtained from A,B,C,D by specialization of the variables to… (More)

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials. The second… (More)

- Mireille Car, Luis H. Gallardo, Olivier Rahavandrainy, Leonid Vaserstein, Leonid Vaserstein
- 2008

Let p be a prime number. It is known that the order o(r) of a root r of the irreducible polynomial xp−x−1 over Fp divides g(p) = p p−1 p−1 . Samuel Wagstaff recently conjectured that o(r) = g(p) for any prime p. The main object of the paper is to give some subsets S of {1, . . . , g(p)} that do not contain o(r).

It is well known that every l∞ linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to an l∞ linear approximation problem. Now we recall relevent definitions. An affine function of variables x1, . . . , xn is b0 + c1x1 + · · · + cnxn where b0, ci are given numbers. A… (More)

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.

with unknown xi. In 1769 Euler conjectured that (1.1) has no positive integral solutions when m < n. This generalizes both his result for n1⁄4 3 and Fermat’s Last Theorem. However his conjecture was refuted for mþ 11⁄4 n1⁄4 5 (Lander and Parkin, 1966) and then for mþ 11⁄4 n1⁄4 4 (Elkies, 1988). It is still unknown whether (1.1) has solutions with mþ 11⁄4 n… (More)

Let A be a local commutative principal ideal ring. We study the double coset space of GLn(A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A. We introduce some invariants of the double cosets. If k is the length of the ring, we determine… (More)

- Luis H. Gallardo, Olivier Rahavandrainy, Leonid Vaserstein
- Finite Fields and Their Applications
- 2007

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