Square-free values of polynomials over the rational function field

JOURNAL OF NUMBER THEORY

卷:135

Article

Rudnick, Zeev

关键词: Square-free;    Finite fields;    Function fields;   

DOI  :  10.1016/j.jnt.2013.08.014

来源: Elsevier

【 摘 要 】

We study representation of square-free polynomials in the polynomial ring F-q[t] over a finite field F-q by polynomials in F-q[t][x]. This is a function field version of the well-studied problem of representing square-free integers by integer polynomials, where it is conjectured that a separable polynomial f is an element of Z[w] takes infinitely many square-free values, barring some simple exceptional cases, in fact that the integers a for which f (a) is square-free have a positive density. We show that if f(x) E is an element of F-q [t] [x] is separable, with square-free content, of bounded degree and height, and n is fixed, then as q -> infinity, for almost all monic polynomials a(t) of degree n, the polynomial f(a) is square-free. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free