A Polynomial Sieve and Sums of Deligne Type

Let $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X_0,\mathrm{.}\mathrm{.}\mathrm{.},X_n]$ 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 $$N(f,F,B):=\mathrm{\mid }\{\mathbf{\text{x}}\in {\mathbb{Z}}^{n+1}:\underset{0\le i\le n}{\max }\mathrm{\mid }{x}_i\mathrm{\mid }\le B,\mathrm{\exists }t\in \mathbb{Z}\text{ such that }f(t)=F(\mathbf{\text{x}})\}\mathrm{\mid },$$ 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.