Semi-interlaced polytopes

TL;DR

The paper introduces a combinatorial formula for the mixed volume of semi-interlaced polytopes, broadening the class of polytopes for which mixed volumes can be efficiently computed, with applications in algebraic geometry and polynomial system complexity.

Contribution

It extends the concept of interlaced polytopes to semi-interlaced ones, providing a new formula for their mixed volume and applications to algebraic degrees and Milnor number dependence.

Findings

Proved a combinatorial formula for semi-interlaced polytopes' mixed volume.

Included off-coordinate polytopes used in algebraic degree computations.

Applied results to the Arnold monotonicity problem.

Abstract

The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face $F \subsetneq P$ intersects at least $\dim(F) + 1$ of the polytopes $D_i$. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems. Motivated by relaxing the bound $\dim(F) + 1$ to $\dim(F)$, we prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees -- such as Maximum Likelihood, Euclidean Distance, and Polar degrees -- via the Kouchnirenko--Bernshtein theory. We also present applications of our results to the Arnold monotonicity problem (1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra.