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DENSITY OF POWER-FREE VALUES OF POLYNOMIALS

Part of: Multiplicative number theory Diophantine equations

Published online by Cambridge University Press:  14 August 2019

Kostadinka Lapkova  and
Stanley Yao Xiao
Kostadinka Lapkova
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, 8010 Graz, Austria email lapkova@math.tugraz.at
Stanley Yao Xiao
Affiliation:
Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George St., Room 6290, Toronto, Ontario, CanadaM5S 2E4 email stanley.xiao@math.toronto.edu
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Abstract

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

MSC classification

Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values
Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1038 - 1050
Copyright
Copyright © University College London 2019 

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