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1 June 2003 Squarefree values of multivariable polynomials
Bjorn Poonen
Duke Math. J. 118(2): 353-373 (1 June 2003). DOI: 10.1215/S0012-7094-03-11826-8

Abstract

Given $f\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute the density of $x\in \mathbf {Z}\sp n$ such that $f(x)$ is squarefree, assuming the abc-conjecture. Given $f,g\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute unconditionally the density of $x\in \mathbf {Z}\sp n$ such that $\gcd(f(x),g(x))=1$. Function field analogues of both results are proved unconditionally. Finally, assuming the abc-conjecture, given $f\in \mathbf {Z}[x]$, we estimate the size of the image of $f(\{1,2,\ldots n\})$ in $(\mathbf {Q}\sp \ast/\mathbf {Q}\sp {\ast 2})\cup \{0\}$.

Citation

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Bjorn Poonen. "Squarefree values of multivariable polynomials." Duke Math. J. 118 (2) 353 - 373, 1 June 2003. https://doi.org/10.1215/S0012-7094-03-11826-8

Information

Published: 1 June 2003
First available in Project Euclid: 23 April 2004

zbMATH: 1047.11021
MathSciNet: MR1980998
Digital Object Identifier: 10.1215/S0012-7094-03-11826-8

Subjects:
Primary: 11Bxx
Secondary: 11Cxx

Rights: Copyright © 2003 Duke University Press

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Vol.118 • No. 2 • 1 June 2003
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