Squarefree values of multivariable polynomials

Abstract

Given f ∈ Z[x1,... ,xn], we compute the density of x ∈ Zn such that f(x) is squarefree, assuming the abc-conjecture. Given f,g ∈ Z[x1,...,xn], we compute unconditionally the density of x ∈ 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 ∈ Z[x], we estimate the size of the image of f({1,2,..., n}) in (Q*/Q*2) ∪ {0}.