A Polynomial Sieve and Sums of Deligne Type

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 present a new bound for $N(f,F,B):=|\{\textbf{x}\in \mathbb{Z}^{n+1}:\max-{0\leq i\leq n}|x-{i}|\leq B,\exists t\in \mathbb{Z}\textrm{ such that}\ f(t)=F(\textbf{x})\}|.$ To do this, we introduce a generalization of the power sieve [14, 28] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.