A polynomial generalization of the Euler characteristic for algebraic sets

TL;DR

This paper introduces a polynomial-based method for computing the Euler characteristic of algebraic sets in complex space, utilizing classical algebraic tools, and proposes a new invariant linked to counting rational points over finite fields.

Contribution

It presents a novel polynomial generalization of the Euler characteristic for algebraic sets, enabling new ways to analyze their properties and relate to finite field point counting.

Findings

Provides a computational method using Gröbner bases and primary decomposition.

Defines a new invariant for algebraic varieties based on this method.

Establishes a connection between the invariant and rational point counting over finite fields.

Abstract

We present a method to compute the Euler characteristic of an algebraic subset of $\bc^n$. This method relies on clasical tools such as Gr\"obner basis and primary decomposition. The existence of this method allows us to define a new invariant for such varieties. This invariant is related to the poblem of counting rational points over finite fields.