Squarefree values of multivariable polynomials

Details Squarefree values of multivariable polynomials Squarefree values of multivariable polynomials

2002-03-29

arxiv.org/abs/math/0203292v2

Mathematics

Number Theory

16 pages, Latex 2e, will appear in Duke Mathematical Journal

Scientific paper

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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