| Résume | For $n>1$, consider an absolutely irreducible polynomial $F(Y,X_1,...,X_n)$ that is a polynomial in $Y^m$ 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 $N(F,B) \ll_{\epsilon} log(||F||) ^c B^{n-1+1/(n+1)+\epsilon}$ under a non-degeneracy condition that encapsulates that $F(Y,X_1,...,X_n)$ is truly a polynomial in $n+1$ variables. A strength of this result is that it requires no smoothness assumptions for $F(Y,X_1,...,X_n)$ nor constraints on the degrees of $F$ in $X_1,...,X_n$. 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. |
