Moscow Mathematical Journal

Volume 26, Issue 2, April–June 2026 pp. 167–187.

Semi-Interlaced Polytopes

Authors: Fedor Selyanin (1)
Author institution:(1) Skolkovo Institute of Science and Technology, Moscow, Russia;
Department of Mathematics, National Research University “Higher School of Economics”, Moscow, Russia;
International Laboratory of Cluster Geometry, National Research University “Higher School of Economics”, Moscow, Russia

Summary:

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 $n! \operatorname{Vol}(P)$. 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 (Problem 1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra.

2020 Math. Subj. Class. 52A39, 14M25.

Keywords: Mixed volume, Newton polytope, algebraic degrees, projective toric variety.

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