Comptage des multiplicit\'es dans une hypersurface sur un corps fini

TL;DR

This paper establishes an upper bound on the sum of multiplicities of rational points on hypersurfaces over finite fields, linking geometric complexity to algebraic properties using intersection trees.

Contribution

It introduces the intersection tree concept to bound multiplicities and provides a new upper bound related to hypersurface degree, dimension, and finite field size.

Findings

Upper bound for multiplicity sum in hypersurfaces

Introduction of intersection trees for multiplicity analysis

Description of singular locus complexity

Abstract

In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called intersection tree by intersection theory. We construct a sequence of intersections, such that the multiplicity of a singular rational point is equal to that of one of the irreducible components in these intersections. The multiplicities of these irreducible components constructed above are bounded by their multiplicities in the intersection tree.