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Squarefree values of multivariable polynomials

Bjorn Poonen
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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}.

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Bjorn Poonen. 2002-03-29. Squarefree values of multivariable polynomials. https://doi.org/10.1215/s0012-7094-03-11826-8

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