Computing mixed multiplicities, mixed volumes and sectional Milnor numbers

This is an expository version of our paper [J. Softw. AlgebraGeom., 13 (2023), 1–12]. Our aim is to present recent Macaulay 2 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 [Proc. Amer. Math. Soc. 147(2019), 4605–4616]. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hyper surfaces with an isolated singularity.