Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers

TL;DR

This paper introduces a computational package for calculating mixed multiplicities, volumes, and Milnor numbers, leveraging algebraic algorithms based on multi-Rees algebra equations, applicable to ideals in Noetherian rings and convex polytopes.

Contribution

The paper presents a novel software package that computes mixed multiplicities, volumes, and Milnor numbers using algebraic algorithms derived from generalized Rees algebra equations.

Findings

Successfully computes mixed multiplicities of ideals.

Calculates mixed volumes of convex lattice polytopes.

Determines sectional Milnor numbers of hypersurfaces.

Abstract

We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity. The algorithms make use of the defining equations of the multi-Rees algebra of ideals, which are obtained by generalising a result of D. Cox. K.-N. Lin, and G. Sosa.