Squarefree values of multivariable polynomials
Number Theory
2017-04-03 v2
Abstract
Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^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 Z[x], we estimate the size of the image of f({1,2,...,n}) in (Q^*/Q^*2) union {0}.
Keywords
Cite
@article{arxiv.math/0203292,
title = {Squarefree values of multivariable polynomials},
author = {Bjorn Poonen},
journal= {arXiv preprint arXiv:math/0203292},
year = {2017}
}
Comments
16 pages, Latex 2e, will appear in Duke Mathematical Journal