On exponential sums

TL;DR

This paper establishes bounds for exponential sums over finite fields involving polynomials with singular projective hypersurfaces, using cohomological methods and Milnor numbers to quantify the bounds.

Contribution

It provides new bounds for exponential sums when the highest degree form of the polynomial defines a singular hypersurface, extending previous results to singular cases.

Findings

Bounds involve Milnor numbers of singularities

Applicable to hypersurfaces with line arrangements in P^2

Uses cohomological interpretation and Grothendieck's trace formula

Abstract

Let f be a polinomial with coefficients in a finite field F. Let $\Psi : F \to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds for the exponential sums $\sum_{x\in F^n} \Psi (Tr_{F/F_p} (f(x)))$ in some cases where the highest degree form of f defines a singular projective hypersurface X (e.g. when X is an arrangement of lines in P^2). The bound involves the Milnor numbers of the singularities of X. The proof goes via the classical cohomological interpretation of this exponential sums through Grothendieck's trace formula.