Counting integral points in thin sets of type II: singularities, sieves, and stratification

Consider an absolutely irreducible polynomial $F(Y,X_1,\ldots,X_n) \in \mathbb{Z}[Y,X_1,\ldots,X_n]$ that is monic in $Y$ and is a polynomial in $Y^m$ for an integer $m \geq 1$. Let $N(F,B)$ count the number of $\mathbf{x} \in [-B,B]^n \cap \mathbb{Z}^n$ such that $F(y,\mathbf{x})=0$ is solvable for $y \in\mathbb{Z}$. In nomenclature of Serre, bounding $N(F,B)$ corresponds to counting integral points in an affine thin set of type II. Previously, in this generality Serre proved $N(F,B) \ll_F B^{n-1/2}(\log B)^{\gamma}$ for some $\gamma<1$. When $m \geq 2$, this new work proves $N(F,B) \ll_{n,F,\epsilon} B^{n-1+1/(n+1) + \epsilon}$ under a nondegeneracy condition that encapsulates that $F(Y,\mathbf{X})$ is truly a polynomial in $n+1$ variables, even after performing any $\text{GL}_n(\mathbb{Q})$ change of variables on $X_1,\ldots,X_n$. Under GRH, this result also holds when $m=1$. We show that generic polynomials satisfy the relevant nondegeneracy condition. Moreover, for a certain class of polynomials, we prove the stronger bound $N(F,B) \ll_{F} B^{n-1}(\log B)^{e(n)}$, comparable to a conjecture of Serre. A key strength of these results is that they require no nonsingularity property of $F(Y,\mathbf{X})$. The Katz-Laumon stratification for character sums, in a new uniform formulation appearing in a companion paper of Bonolis, Kowalski and Woo, is a key ingredient in the sieve method we develop to prove upper bounds that explicitly control any dependence on the size of the coefficients of $F$.