  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.
