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The distance to square-free polynomials

Volume 186 / 2018

Artūras Dubickas, Min Sha Acta Arithmetica 186 (2018), 243-256 MSC: 11C08, 11T06. DOI: 10.4064/aa180122-15-8 Published online: 19 October 2018


We consider a variant of Turán’s problem on the distance from a polynomial in $\mathbb Z[x]$ to the nearest irreducible polynomial in $\mathbb Z[x]$. We prove that for any $f \in \mathbb Z[x]$, there exist infinitely many square-free polynomials $g\in \mathbb Z[x]$ such that ${L(f-g) \le 2}$, 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 $L(f-g) \le 1$. For this, for each integer $d \geq 15$ we construct infinitely many polynomials $f \in \mathbb Z[x]$ of degree $d$ such that neither $f$ itself nor any $f(x) \pm x^k$, where $k$ is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.


  • Artūras DubickasInstitute of Mathematics
    Faculty of Mathematics and Informatics
    Vilnius University
    Naugarduko 24
    Vilnius LT-03225, Lithuania
  • Min ShaDepartment of Computing
    Macquarie University
    Sydney, NSW 2109, Australia

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