Density of power-free values of polynomials

Abstract

We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots ,x_n)\in \mathbb{Z}[x_1,\cdots ,x_n]$ of degree $d\ge 2$ for any $n\ge 1$, including when the variables are prime, as long as $k\ge (3d+1)/4$. Thus we generalize a work of Browning, while we use a different sieveing technique for the middle range of primes.

12 pages, comments welcome

References in corpus (3)

• Squarefree values of polynomial discriminants I
• Density of power-free values of polynomials
• Square-free values of decomposable forms

Cited by in corpus (2)

• Density of power-free values of polynomials
• A stratification result for an exponential sum modulo $p^2$