Arnold's monotonicity problem
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2020
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913479316733952 |
|---|---|
| author | Selyanin, Fedor |
| author_facet | Selyanin, Fedor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_11795 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Arnold's monotonicity problem Selyanin, Fedor Algebraic Geometry 14B05, 14B07, 14D05, 14D06, 05E45, 52B20 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. |
| title | Arnold's monotonicity problem |
| topic | Algebraic Geometry 14B05, 14B07, 14D05, 14D06, 05E45, 52B20 |
| url | https://arxiv.org/abs/2006.11795 |