Square-free values of multivariate polynomials over function fields in linear sparse sets

TL;DR

This paper investigates the occurrence of square-free values of multivariate polynomials over function fields within linear sparse sets, establishing conditions under which such values are abundant as the field size grows.

Contribution

It extends the understanding of square-free values of polynomials over function fields to multivariate cases and sparse linear subsets, providing new existence results under large field conditions.

Findings

Existence of square-free polynomial values in sparse linear sets for large q.

Almost all parameter choices yield square-free values as q increases.

Generalization of results to multivariate polynomials.

Abstract

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial f in Z[x] takes infinitely many square-free values. Let c(t) be a polynomial in Fq[t] of degree less than m, and let k < m be coprime to q. A consequence of the main result we show, is that if q is sufficiently large with respect to m and the degrees of f in t and x, then there exist $\beta_1,\beta_2$ in Fq such that $f(t,c(t)+\beta_1t^k+\beta_2)$ is square-free. Moreover, as q tends to infinity, the last is true for almost all $\beta_1$ and $\beta_2$ in Fq. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.