The multivariate arithmetic Tutte polynomial

Petter Brändèn; Luca Moci

doi:10.46298/dmtcs.3072

Petter Brändèn ; Luca Moci - The multivariate arithmetic Tutte polynomial

dmtcs:3072 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3072

The multivariate arithmetic Tutte polynomialConference paper

Authors: Petter Brändèn ¹; Luca Moci ²

1 Department of Mathematics [Stockholm
2 Department of Mathematics [Roma]

[en]
We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.

[fr]
Nous introduisons une version arithmétique du polynôme de Tutte multivariée récemment étudié par Sokal, et un quasi-polynôme qui interpole entre les deux. Nous proposons une représentation de Fortuin-Kasteleyn neutralise pour les matroïdes arithmétiques représentables, avec des applications aux colorations et flux arithmétiques. Nous donnons une nouvelle preuve de la positivité des coefficients du polynôme de Tutte arithmétique dans le cadre plus général des matroïdes pseudo-arithmétiques. Dans le cas d'un matroïde arithmétique représentable, nous proposons une interprétation géométrique des coefficients du polynôme de Tutte arithmétique.

https://doi.org/10.46298/dmtcs.3072

Source: HAL:hal-01283095v1

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)

Section: Proceedings

Published on: January 1, 2012

Imported on: January 31, 2017

Keywords: [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [en] Potts model, Tutte polynomial, chromatic polynomial, matroids, arithmetic matroids, abelian groups.

Bibliographic References

21 Documents citing this article

Elia Saini, 2024, Centering toric arrangements of maximal rank, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 23, 1, pp. 39-46, 10.2478/aupcsm-2024-0005, https://doi.org/10.2478/aupcsm-2024-0005.

Tianlong Ma;Xian'an Jin;Weiling Yang, 2024, Convolution formulas for multivariate arithmetic Tutte polynomials, Advances in Applied Mathematics, 157, pp. 102692, 10.1016/j.aam.2024.102692.

Akihiro Higashitani;Tan Nhat Tran;Masahiko Yoshinaga, 2022, Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements, International Mathematics Research Notices, 2023, 10, pp. 8934-8963, 10.1093/imrn/rnac104.

Luca Moci;Roberto Pagaria, 2022, On the cohomology of arrangements of subtori, Journal of the London Mathematical Society, 106, 3, pp. 1999-2029, 10.1112/jlms.12616, https://doi.org/10.1112/jlms.12616.

Emanuele Delucchi;Roberto Pagaria;Emanuele Delucchi;Roberto Pagaria, 2021, The homotopy type of elliptic arrangements, arXiv (Cornell University), 21, 4, pp. 2037-2063, 10.2140/agt.2021.21.2037, http://arxiv.org/abs/1911.02905.

Ahmed Umer Ashraf;Tan Nhat Tran;Masahiko Yoshinaga;Ahmed Umer Ashraf;Tan Nhat Tran;et al., 2020, Eulerian polynomials for subarrangements of Weyl arrangements, arXiv (Cornell University), 120, pp. 102064, 10.1016/j.aam.2020.102064, http://arxiv.org/abs/1911.01650.

Roberto Pagaria, 2020, Orientable arithmetic matroids, arXiv (Cornell University), 343, 6, pp. 111872, 10.1016/j.disc.2020.111872, http://arxiv.org/abs/1805.11888.

Roberto Pagaria;Giovanni Paolini, 2020, Representations of torsion-free arithmetic matroids, arXiv (Cornell University), 93, pp. 103272, 10.1016/j.ejc.2020.103272, http://arxiv.org/abs/1908.04137.

Tan Nhat Tran, 2020, An equivalent formulation of chromatic quasi-polynomials, arXiv (Cornell University), 343, 10, pp. 112012, 10.1016/j.disc.2020.112012, http://arxiv.org/abs/1803.08649.

Matthias Lenz, 2019, On powers of Plücker coordinates and representability of arithmetic matroids, arXiv (Cornell University), 112, pp. 101911, 10.1016/j.aam.2019.04.008, http://arxiv.org/abs/1703.10520.

Matthias Lenz;Matthias Lenz, 2019, Representations of Weakly Multiplicative Arithmetic Matroids are Unique, arXiv (Cornell University), 23, 2, pp. 335-346, 10.1007/s00026-019-00424-z, http://arxiv.org/abs/1704.08607.

Roberto Pagaria, 2019, Poincaré polynomial of elliptic arrangements is not determined by the Tutte polynomial, arXiv (Cornell University), 343, 6, pp. 111710, 10.1016/j.disc.2019.111710, http://arxiv.org/abs/1810.06248.

Roberto Pagaria, 2019, Two examples of toric arrangements, arXiv (Cornell University), 167, pp. 389-402, 10.1016/j.jcta.2019.05.006, http://arxiv.org/abs/1804.05767.

Tan Nhat Tran;Masahiko Yoshinaga, 2019, Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials, arXiv (Cornell University), 165, pp. 258-272, 10.1016/j.jcta.2019.02.003, http://arxiv.org/abs/1805.03365.

Ye Liu;Tan Nhat Tran;Masahiko Yoshinaga;Ye Liu;Tan Nhat Tran;et al., 2019, G-Tutte Polynomials and Abelian Lie Group Arrangements, International Mathematics Research Notices, 2021, 1, pp. 150-188, 10.1093/imrn/rnz092, https://doi.org/10.1093/imrn/rnz092.

Alex Fink;Luca Moci, 2018, Polyhedra and parameter spaces for matroids over valuation rings, arXiv (Cornell University), 343, pp. 448-494, 10.1016/j.aim.2018.11.009, http://arxiv.org/abs/1707.01026.

Emanuele Delucchi;Luca Moci, 2018, Products of arithmetic matroids and quasipolynomial invariants of CW-complexes, Archivio istituzionale della ricerca (Alma Mater Studiorum Università di Bologna), 157, pp. 28-40, 10.1016/j.jcta.2018.01.005, http://hdl.handle.net/11585/676940.

R. B. Bapat;Sivaramakrishnan Sivasubramanian, 2018, The Arithmetic Tutte polynomial of two matrices associated to Trees, Special Matrices, 6, 1, pp. 310-322, 10.1515/spma-2018-0026, https://doi.org/10.1515/spma-2018-0026.

Alex Fink;Luca Moci, 2015, Matroids Over a Ring, Springer INdAM series, pp. 41-47, 10.1007/978-3-319-20155-9_9.

Francesco Cavazzani;Luca Moci, 2015, Geometric Realizations and Duality for Dahmen–Micchelli Modules and De Concini–Procesi–Vergne Modules, Discrete & Computational Geometry, 55, 1, pp. 74-99, 10.1007/s00454-015-9745-3.

Matthias Lenz, 2015, Splines, lattice points, and arithmetic matroids, arXiv (Cornell University), 43, 2, pp. 277-324, 10.1007/s10801-015-0621-2, http://arxiv.org/abs/1408.4041.

Sources : OpenCitations, OpenAlex & Crossref

Share and export

Consultation statistics

This page has been seen 416 times.

This article's PDF has been downloaded 635 times.