P. B. Bhattacharya and J. J. Risler - B. Teissier proved that if $I_0,I_1,\ldots,I_r$ are $m$-primary ideals in a Noetherian local ring $(R,m)$ of dimension $d$, then the function $B(u_0,u_1,\ldots,u_r) = l(R/I_0^{u_0}I_1^{u_1} \cdots I_r^{u_r})$ is a polynomial function in $u_0,u_1,\ldots,u_r$ of degree $d$ for large values of $u_0,u_1,\ldots,u_r.$ The coefficients of the top degree term are called the mixed multiplicities of the ideals $I_0,I_1,\ldots,I_r.$ This result was generalized for ideals of positive height in the works of D. Katz, J. K. Verma and N. V. Trung. D. Rees studied these numbers using complete and joint reductions of ideals.
Our algorithm to compute the mixed multiplicities requires computation of the defining ideal of the multi-Rees algebra of ideals. An expression of the defining ideal of the multi-Rees algebra of monomial ideals over a polynomial ring was given by D. Cox, K.-i. Lin and G. Sosa in (Multi-Rees algebras and toric dynamical systems. Proc. Amer. Math. Soc., 147(11):4605-4616, 2019). We use a generalization of their result to compute the defining ideal of multi-Rees algebras of ideals over domains. Otherwise, we use a generalization of the algorithm of the function reesIdeal to the multi-ideal case.
The computation of mixed multiplicities helps compute mixed volume of a collection of lattice polytopes and also the sectional Milnor numbers of hypersurfaces with an isolated singularity.
Let $Q_1,\ldots,Q_n$ be a collection of lattice polytopes in $\mathbb{R}^n$ and $t_1,\ldots,t_n \in \mathbb{R}_+$. Minkowski proved that the $n$-dimensional volume, $vol_n(t_1Q_1 + \cdots + t_nQ_n)$ is a homogeneous polynomial of degree $n$ in $t_1,\ldots,t_n.$ The coefficient of $t_1 \cdots t_n$ is called the mixed volume of $Q_1,\ldots,Q_n.$ N. V. Trung and J. K. Verma proved that the mixed volume of lattice polytopes in the above setup can be realized as a mixed multiplicity of the homogeneous ideals corresponding to the lattice polytopes.
Let origin be an isolated singular point of a complex analytic hypersurface $H = V(f).$ The $\mathbb{C}$-vector space dimension of $\mathbb{C}\{x_0,\ldots,x_n\}/(f_{x_0},\ldots,f_{x_n})$ is called the Milnor number of the hypersurface $H$ at the origin. Let $(X, 0)$ be a germ of a hypersurface in $\mathbb{C}^{n+1}$ with an isolated singularity. The Milnor number of $X \cap E$, where $E$ is a general linear subspace of dimension $i$ passing through $0,$ is called the $i$-th sectional Milnor number of $X$. B. Teissier identified the $i$-th sectional Milnor number with the $i$-th mixed multiplicity of the maximal homogeneous ideal of the polynomial ring and the Jacobian ideal of $f.$
Version 3.0 of this package was accepted for publication in volume 13 of Journal of Software for Algebra and Geometry on 2023-06-06, in the article Algorithms for computing mixed multiplicities, mixed volumes, and sectional Milnor numbers. That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 3.0 of MixedMultiplicity.
The source code from which this documentation is derived is in the file MixedMultiplicity.m2.
The object MixedMultiplicity is a package.