Kronecker conjugacy of polynomials

Kronecker Conjugacy of Polynomials

Let f, g ∈ Z[X] be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets f(Z) and g(Z) are the same modulo all but finitely many primes. The main progress until now is M. Fried’s result that f and g then differ by a linear substitution, provided that f is functionally indecomposable. We extend this result to polynomials f of comp...

An Infinite Series of Kronecker Conjugate Polynomials

Let K be a field of characteristic 0, t a transcendental over K, and Γ be the absolute Galois group of K(t). Then two non-constant polynomials f, g ∈ K[X] are said to be Kronecker conjugate if an element of Γ fixes a root of f(X) − t if and only if it fixes a root of g(X) − t. If K is a number field, and f, g ∈ OK [X] where OK is the ring of integers of K, then f and g are Kronecker conjugate i...

Character Polynomials, their q-Analogs and the Kronecker Product

Dedicated to Anders Björner on the occasion of his sixtieth birthday. Abstract The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomi-als. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an " umbral " ver...

Sparse Interpolation of Black-box Multivariate Polynomials using Kronecker Type Substitutions

In this paper, we give two new deterministic interpolation algorithms for black-box multivariate polynomials. For a t-sparse and n-variate polynomial f with a degree bound D and a term bound T , we show that at least half of the terms of f can be recovered from the univariate polynomials obtained from f by three types of Kronecker substitutions. We then give a criterion to check whether these t...

Solutions to a family of matrix equations by using the Kronecker matrix polynomials