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

2014/10/27 by Shai Rosenberg, Rosenberg, Shai
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.1410.7449

Moved one section to the appendix, and made minor edits

openalex publication_date 2014/10/27 · arxiv created 2015/03/03 · arxiv updated 2015/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 β₁,β₂ in Fq such that f(t,c(t)+β₁t^k+β₂) is square-free. Moreover, as q tends to infinity, the last is true for almost all β₁ and β₂ in Fq. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.

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