Semi-interlaced polytopes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914563293708288 |
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| author | Selyanin, Fedor |
| author_facet | Selyanin, Fedor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13410 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semi-interlaced polytopes Selyanin, Fedor Combinatorics Algebraic Geometry 52A39, 14M25 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. |
| title | Semi-interlaced polytopes |
| topic | Combinatorics Algebraic Geometry 52A39, 14M25 |
| url | https://arxiv.org/abs/2605.13410 |