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

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.