Density of m-tuples with respect to polynomials (Q1089040)

The authors obtain asymptotic formulas for the number of m-tuples \(<x_ 1,...,x_ m>\) of integers satisfying \(0<x_ i\leq x\) and with a given value of the greatest k-th power dividing \(f_ i(x_ i)\) for \(i=1,2,...,m\), where \(f_ 1,...,f_ m\) are given nonconstant polynomials with integer coefficients. They also point out errors in a previous treatment of this subject [\textit{R. N. Buttsworth}, J. Number Theory 12, 487-498 (1980; Zbl 0443.10001)].