Arnold's monotonicity problem

TL;DR

This paper develops a non-negative analogue of the Kouchnirenko formula for Milnor numbers, providing new criteria for Arnold's monotonicity problem and solving it in dimensions up to 4 with partial results in dimension 5.

Contribution

It introduces a non-negative formula for the Milnor number, enabling a criterion for Arnold's monotonicity problem in arbitrary dimensions, with complete solutions up to dimension 4.

Findings

Complete solution to Arnold's monotonicity problem in dimensions up to 4

Partial solution in dimension 5 based on classification of thin triangulations

Examples in dimension 4 that differ significantly from lower-dimensional cases

Abstract

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.