Application of a polynomial sieve: beyond separation of variables

Abstract

Let a polynomial $f\in \mathbb{Z}[X_1,\dots <!--FUNCTION\; APPLICATION-->,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $x\in {[-B,B]}^n$ such that $f(x)$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $x\in {[-B,B]}^n$ for which $f(x)=y^r$ is solvable for $y\in \mathbb{Z}$; then to a polynomial sieve, counting $x\in {[-B,B]}^n$ for which $f(x)=g(y)$ is solvable, for a given polynomial $g$. Formally, a polynomial sieve lemma can encompass the more general problem of counting $x\in {[-B,B]}^n$ for which $F(y,x)=0$ is solvable, for a given polynomial $F$. Previous applications, however, have only succeeded in the case that $F(y,x)$ exhibits separation of variables, that is, $F(y,x)$ takes the form $f(x)-g(y)$. In the present work, we present the first application of a polynomial sieve to count $x\in {[-B,B]}^n$ such that $F(y,x)=0$ 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.