The distance to square-free polynomials

Artūras Dubickas, Min Sha

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We consider a variant of Turán’s problem on the distance from a polynomial in Z[x] to the nearest irreducible polynomial in Z[x]. We prove that for any f∈Z[x], there exist infinitely many square-free polynomials g∈Z[x] such that L(f−g)≤2, where L(f−g) denotes the sum of the absolute values of the coefficients of f−g. On the other hand, we show that this inequality cannot be replaced by L(f−g)≤1. For this, for each integer d≥15 we construct infinitely many polynomials f∈Z[x] of degree d such that neither f itself nor any f(x)±xk, where k is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.
Original languageEnglish
Pages (from-to)243-256
Number of pages14
JournalActa Arithmetica
Volume186
Issue number3
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Integer polynomial
  • Square-free polynomial
  • Turán’s problem

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