Computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

TL;DR

This paper presents recent algorithms implemented in Macaulay2 for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers, enhancing efficiency and generality in algebraic and geometric computations.

Contribution

It introduces new algorithms based on multi-Rees algebra equations for computing mixed multiplicities, volumes, and Milnor numbers, extending previous methods.

Findings

Algorithms successfully compute mixed multiplicities in Noetherian rings.

Efficient computation of mixed volumes of convex lattice polytopes.

Calculation of sectional Milnor numbers for hypersurfaces with isolated singularities.

Abstract

This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that generalises a result of Cox, Lin and Sosa. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity.