The distance to square-free polynomials

Abstract:

Given an integer polynomial ff, let L(f)L(f) be the sum of the absolute values of the coefficients of ff. In 1960s, Turán asked whether there exists an absolute constant CC such that for any integer polynomial ff of degree dd,  there is an irreducible integer polynomial gg of degree at most d satisfying L(f−g)<CL(f−g)<C. Turán's problem remains open, although a number of partial results have been obtained.

In this talk, I will present some recent work on a variant of Turán's problem. For example, we prove that for any integer polynomial ff, there exist infinitely many square-free integer polynomials gg such that L(f−g)<3L(f−g)<3. On the other hand, we show that this inequality cannot be replaced by L(f−g)<2L(f−g)<2. (This is joint work with Artūras Dubickas)