arXiv subjects

Fedor Selyanin

Publications and source records attributed to Fedor Selyanin.

4 recordsLinked to original sources

arXiv2026

Semi-interlaced polytopes

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.

math.CO↗

arXiv2025

Newton numbers, vanishing polytopes and algebraic degrees

Consider a polynomial $f$ with a convenient Newton polytope $P$ and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface $\{f = 0\} \subset \mathbb{C}^n$ has the homotopy type of a bouquet of $(n-1)$-spheres, and the number of spheres is given by a certain alternating sum of volumes, called the Newton number $ν(P)$. Using the Furukawa-Ito classification of dual defective sets, we classify convenient Newton polytopes with vanishing Newton numbers as certain Cayley sums called $B_k$-polytopes. These $B_k$-polytopes generalize the $B_1$- and $B_2$-facets appearing in the local monodromy conjecture in the Newton non-degenerate case. Our classification provides a partial solution to Arnold's monotonicity problem. The local $h^*$-polynomial (or $\ell^*$-polynomial) is a natural invariant of lattice polytopes that refines the $h^*$-polynomial coming from Ehrhart theory. We obtain decomposition formulas for the Newton number, for instance, prove the inequality $ν(P) \ge \ell^*(P;1)$. The $B_k$-polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the $\ell$-Newton number and the $e$-Newton number. The $\ell$-Newton number comes from Ehrhart theory, namely, from certain generalizations of Katz-Stapledon decomposition formulas, and its properties are central to our proof that the $B_k$-polytopes are thin. The $e$-Newton number is the number of points of zero-dimensional critical complete intersections. Vanishing of the $e$-Newton number characterizes dual defective sets. Furthermore, the $e$-Newton number calculates algebraic degrees (such as Maximum Likelihood, Euclidean Distance and Polar degrees). For instance, we show that all known formulas for these algebraic degrees in the Newton non-degenerate case are implied by basic properties of the $e$-Newton number.

math.CO↗

arXiv2025

B-facets in dimension 4

We complete the classification of B-facets of a 4-dimensional Newton polyhedron, filling a gap in the classification of arXiv:1309.0630, found by the authors of arXiv:2209.03553.

math.CO↗

arXiv2024

Arnold's monotonicity problem

According to the Kouchnirenko formula, the Milnor number of a generic isolated singularity with given Newton polyhedron is equal to the alternating sum of certain volumes associated to the Newton polyhedron. In this paper we obtain a non-negative analogue (i.e. without negative summands) of the Kouchnirenko formula. The analogue relies on the non-negative formula for the monodromy operator from arXiv:1405.5355 and formulas for the Milnor number from arXiv:math/9901107 . As an application we give a criterion for the Arnold's monotonicity problem (1982-16) in arbitrary dimension, which leads to complete solution in dimension up to $4$ and partial solution in dimension $5$. The latter relies on the classification of thin triangulations (or vanishing local h-polynomial) in dimension $2$ and $3$ from arXiv:1909.10843 (and from the book by Gelfand, Kapranov and Zelevinsky) and contains examples which differ dramatically from the ones which arise in dimension up to $3$ in arXiv:1705.00323 (see also arXiv:2001.10316 ). Some of the $4$-dimensional examples were first described in arXiv:1309.0630 in the context of the local monodromy conjecture.

math.AG↗