On intersective polynomials with non-solvable Galois group

TL;DR

This paper investigates the existence of intersective polynomials with specific non-solvable Galois groups, providing new theoretical results and explicit examples for groups like $PGL_2(\, ext{ell})$, $AGL_2(\, ext{ell})$, and $AGSp_4(\, ext{ell})$, advancing understanding in Galois theory.

Contribution

It introduces new theoretical results on intersective polynomials with certain non-solvable Galois groups and computes explicit examples for some of these groups.

Findings

Existence results for polynomials with $PGL_2(\, ext{ell})$, $AGL_2(\, ext{ell})$, and $AGSp_4(\, ext{ell})$ groups.

Conditional existence results for families of affine groups based on tamely ramified extensions.

Explicit families of intersective polynomials for specific non-solvable groups.

Abstract

We present new theoretical results on the existence of intersective polynomials with certain prescribed Galois groups, namely the projective and affine linear groups $PGL_2(\ell)$ and $AGL_2(\ell)$ as well as the affine symplectic groups $AGSp_4(\ell):=(\mathbb{F}_\ell)^4 \rtimes GSp_4(\ell)$. For further families of affine groups, existence results are proven conditional on the existence on certain tamely ramified Galois extensions. We also compute explicit families of intersective polynomials for certain non-solvable groups.