Counting Multiplicities in a Hypersurface over a Number Field

TL;DR

This paper establishes an upper bound for the sum of multiplicities of algebraic points on a hypersurface over a number field, considering height, degree, and singular locus dimensions.

Contribution

It introduces a new counting function for multiplicities and provides bounds for sums over algebraic points with bounded height and fixed degree.

Findings

Derived an explicit upper bound for the sum of multiplicities.

Connected the bound to hypersurface degree, singular locus dimension, and height constraints.

Extended counting techniques to algebraic points over number fields.

Abstract

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.