Squarefree density of polynomials
Volume 214 / 2024
Abstract
This paper is concerned with squarefree values of polynomials \[ \mathcal P(\mathbf x) \in \mathbb Z[x_1,\ldots ,x_s] \] where we suppose that for each $j\le s$ we have $|x_j|\le P_j$. Then we define \[ N_{\mathcal P} (\mathbf P) = \sum _{\substack {\mathbf x\\ |x_j|\le P_j\\ \mathcal P(\mathbf x)\not =0}} \mu (|\mathcal P(\mathbf x)|)^2 \] and we are interested in its behaviour when $\min _jP_j\rightarrow \infty $, and the extent to which this can be approximated by \[ N_{\mathcal P} (\mathbf P) \sim 2^sP_1\ldots P_s\mathfrak S_{\mathcal P} \] where \[ \mathfrak S_{\mathcal P} = \prod _p \bigg ( 1-\frac {\rho _{\mathcal P}(p^2)}{p^{2s}} \bigg )\quad \text {and}\quad \rho _{\mathcal P}(d) = \mathrm{card}\,\{\mathbf x\in \mathbb Z_d^s: \mathcal P(\mathbf x) \equiv 0\ ({\rm mod}\ d)\}. \] We establish this in a number of new cases, and in particular show that if $s\ge 2$ and $\mathfrak S_{\mathcal P}=0$, then \[ N_{\mathcal P} (\mathbf P) =o(P_1\ldots P_s) \] as $\min _jP_j\rightarrow \infty $.
