Application of a polynomial sieve: beyond separation of variables

Abstract

Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $\mathbf{x} \in [-B,B]^n$ such that $f(\mathbf{x})$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=y^r$ is solvable for $y \in \mathbb{Z}$; then to a polynomial sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=g(y)$ is solvable, for a given polynomial $g$. Formally, a polynomial sieve lemma can encompass the more general problem of counting $\mathbf{x} \in [-B,B]^n$ for which $F(y,\mathbf{x})=0$ is solvable, for a given polynomial $F$. Previous applications, however, have only succeeded in the case that $F(y,\mathbf{x})$ exhibits separation of variables, that is, $F(y,\mathbf{x})$ takes the form $f(\mathbf{x}) - g(y)$. In the present work, we present the first application of a polynomial sieve to count $\mathbf{x} \in [-B,B]^n$ such that $F(y,\mathbf{x})=0$ is solvable, in a case for which $F$ does not exhibit separation of variables. Consequently, we obtain a new result toward a question of Serre, pertaining to counting points in thin sets.