Applying stratification theorems to counting integral points in thin sets of type II

For n > 1, consider an absolutely irreducible polynomial F(Y, X₁, …, Xₙ) that is a polynomial in Yᵐ and monic in Y. Let N(F, B) be the number of integral vectors x of height at most B such that there is an integral solution to F(Y, x) = 0. For m > 1 unconditionally, and m = 1 under GRH, we show that
N(F, B) ≪_ε log(∥F∥)ᶜ B^(n-1 + 1/(n+1) + ε)

under a non-degeneracy condition that encapsulates that F(Y, X₁, …, Xₙ) is truly a polynomial in n + 1 variables. A strength of this result is that it requires no smoothness assumptions on F(Y, X₁, …, Xₙ), nor any constraints on the degrees of F in X₁, …, Xₙ. A key ingredient in this work is a formulation of the Katz–Laumon stratification theorems for exponential sums that is uniform in families. This talk is based on joint work with Dante Bonolis, Emmanuel Kowalski, and Lillian B. Pierce.