Squarefree values of multivariable polynomials
Bjorn Poonen
Abstract
Given f in Z[x1,...,xn], we compute the density of x in Zn such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x1,...,xn], we compute unconditionally the density of x in Zn 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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