A Polynomial Sieve and Sums of Deligne Type

Abstract

Let $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X_0,...,X_n]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

Theorem 1 has been improved. The paper has been reorganized to improve the exposition