A note on the squarefree density of polynomials

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Abstract

The conjectured squarefree density of an integral polynomial (Formula presented.) in (Formula presented.) variables is an Euler product (Formula presented.) which can be considered as a product of local densities. We show that a necessary and sufficient condition for (Formula presented.) to be 0 when (Formula presented.) is a polynomial in (Formula presented.) variables over the integers, is that either there is a prime (Formula presented.) such that the values of (Formula presented.) at all integer points are divisible by (Formula presented.) or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density (Formula presented.) satisfies (Formula presented.).

Original languageEnglish (US)
Article numbere12275
JournalMathematika
Volume70
Issue number4
DOIs
StatePublished - Oct 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics

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