Relative Node Polynomials for Plane Curves
International audience We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generaliz...
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ftccsdartic:oai:HAL:hal-01215084v1 2023-05-15T16:52:30+02:00 Relative Node Polynomials for Plane Curves Block, Florian Department of Mathematics - University of Michigan University of Michigan Ann Arbor University of Michigan System-University of Michigan System Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://hal.inria.fr/hal-01215084 https://hal.inria.fr/hal-01215084/document https://hal.inria.fr/hal-01215084/file/dmAO0119.pdf en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS hal-01215084 https://hal.inria.fr/hal-01215084 https://hal.inria.fr/hal-01215084/document https://hal.inria.fr/hal-01215084/file/dmAO0119.pdf info:eu-repo/semantics/OpenAccess ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215084 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.199-210 enumerative geometry floor diagram Gromov-Witten theory node polynomial tangency conditions [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] info:eu-repo/semantics/conferenceObject Conference papers 2011 ftccsdartic 2020-12-25T18:15:03Z International audience We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ . Nous généralisons les travaux récents de Fomin et Mikhalkin sur des formules polynomiales pour les degrés de Severi. Le degré de la variété de Severi des courbes planes de degré d et à δ nœuds est donné par un polynôme en d , pour δ fixé et d assez grand. Nous étendons ce résultat aux variétés de Severi généralisées paramétrant les courbes planes et qui, en outre, satisfont à des conditions de tangence d'ordres donnés avec une droite fixée. Nous montrons que les degrés de ces variétés, rééchelonnés de manière appropriée, sont donnés par un ``polynôme de noeud relatif'', défini combinatoirement, en les ordres de tangence, dès que ceux-ci sont assez grands. Nous décrivons une méthode pour calculer ces polynômes pour delta arbitraire, et l'utilisons pour présenter des formules explicites pour δ ≤ 6 . Nous donnons aussi un seuil pour la polynomialité, et calculons les premiers termes dominants pour tout δ . Conference Object Iceland Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Fomin ENVELOPE(39.730,39.730,64.145,64.145) Severi ENVELOPE(27.250,27.250,66.483,66.483) |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
enumerative geometry floor diagram Gromov-Witten theory node polynomial tangency conditions [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
spellingShingle |
enumerative geometry floor diagram Gromov-Witten theory node polynomial tangency conditions [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Block, Florian Relative Node Polynomials for Plane Curves |
topic_facet |
enumerative geometry floor diagram Gromov-Witten theory node polynomial tangency conditions [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ . Nous généralisons les travaux récents de Fomin et Mikhalkin sur des formules polynomiales pour les degrés de Severi. Le degré de la variété de Severi des courbes planes de degré d et à δ nœuds est donné par un polynôme en d , pour δ fixé et d assez grand. Nous étendons ce résultat aux variétés de Severi généralisées paramétrant les courbes planes et qui, en outre, satisfont à des conditions de tangence d'ordres donnés avec une droite fixée. Nous montrons que les degrés de ces variétés, rééchelonnés de manière appropriée, sont donnés par un ``polynôme de noeud relatif'', défini combinatoirement, en les ordres de tangence, dès que ceux-ci sont assez grands. Nous décrivons une méthode pour calculer ces polynômes pour delta arbitraire, et l'utilisons pour présenter des formules explicites pour δ ≤ 6 . Nous donnons aussi un seuil pour la polynomialité, et calculons les premiers termes dominants pour tout δ . |
author2 |
Department of Mathematics - University of Michigan University of Michigan Ann Arbor University of Michigan System-University of Michigan System Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Block, Florian |
author_facet |
Block, Florian |
author_sort |
Block, Florian |
title |
Relative Node Polynomials for Plane Curves |
title_short |
Relative Node Polynomials for Plane Curves |
title_full |
Relative Node Polynomials for Plane Curves |
title_fullStr |
Relative Node Polynomials for Plane Curves |
title_full_unstemmed |
Relative Node Polynomials for Plane Curves |
title_sort |
relative node polynomials for plane curves |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://hal.inria.fr/hal-01215084 https://hal.inria.fr/hal-01215084/document https://hal.inria.fr/hal-01215084/file/dmAO0119.pdf |
op_coverage |
Reykjavik, Iceland |
long_lat |
ENVELOPE(39.730,39.730,64.145,64.145) ENVELOPE(27.250,27.250,66.483,66.483) |
geographic |
Fomin Severi |
geographic_facet |
Fomin Severi |
genre |
Iceland |
genre_facet |
Iceland |
op_source |
ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215084 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.199-210 |
op_relation |
hal-01215084 https://hal.inria.fr/hal-01215084 https://hal.inria.fr/hal-01215084/document https://hal.inria.fr/hal-01215084/file/dmAO0119.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766042841135448064 |