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G-TUTTE POLYNOMIALS VIA COMBINATORICS, TOPOLOGY AND MATROID THEORY
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| Title: | G-TUTTE POLYNOMIALS VIA COMBINATORICS, TOPOLOGY AND MATROID THEORY |
| Other Titles: | G-TUTTE多項式の組み合わせ論的、位相的、マトロイド理論的研究 |
| Authors: | TRAN NHAT TAN Browse this author |
| Issue Date: | 25-Mar-2020 |
| Publisher: | Hokkaido University |
| Abstract: | We introduce and study the notion of G-Tutte polynomial for a list of elements in a finitely generated abelian group and an abelian group G through combinatorial, topological and matroid theoretical aspects. The G-Tutte polynomial establishes a common generalization of several “Tutte-like” polynomials appearing in the literature such as the (arithmetic) Tutte polynomial of realizable (arithmetic) matroid, the characteristic quasi-polynomial of integral arrangement, the Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial of a finite CW-complex. Through combinatorial viewpoint, we generalize the characteristic polynomials of hyperplane and toric arrangements to that of abelian Lie group arrangements and in turn give two arrangement theoretic interpretations for every constituent of the chromatic quasi-polynomial. Passing from general to particular consideration, we give several results on the characteristic quasi-polynomials of arrangements arising from root systems in connection with Ehrhart theory, Eulerian polynomial and signed graph. From topological viewpoint, we prove that the semialgebraic and topological Euler characteristics and Poincaré polynomial of a certain abelian Lie group arrangement can be expressed in terms of the associated G-Tutte polynomial, which generalizes many classical formulas. From matroid theoretical viewpoint, we prove that the G-Tutte polynomial, like many of its specializations, possesses deletion-contraction and convolution formulas, but unlike them, the G-Tutte polynomial may have negative coefficients. We propose some ideas and partial answers for finding under what conditions the G-Tutte polynomial has positive coefficients. |
| Conffering University: | 北海道大学 |
| Degree Report Number: | 甲第13900号 |
| Degree Level: | 博士 |
| Degree Discipline: | 理学 |
| Examination Committee Members: | (主査) 教授 吉永 正彦, 教授 秋田 利之, 准教授 シモーナ セッテパネーラ |
| Degree Affiliation: | 理学院（数学専攻） |
| Type: | theses (doctoral) |
| URI: | http://hdl.handle.net/2115/78459 |
| Appears in Collections: | 課程博士 (Doctorate by way of Advanced Course) > 理学院(Graduate School of Science)
学位論文 (Theses) > 博士 （理学）
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