Bouquet algebra of tonic ideals

JOURNAL OF ALGEBRA

卷:512

Article

Petrovic, Sonja¹  Thoma, Apostolos²  Vladoiu, Marius^3,4 

[1] IIT, Dept Appl Math, Chicago, IL 60616 USA

[2] Univ Ioannina, Dept Math, GR-45110 Ioannina, Greece

[3] Univ Bucharest, Fac Math & Comp Sci, Str Acad 14, RO-010014 Bucharest, Romania

[4] Romanian Acad, Simion Stoilow Inst Math, Res Grp Project PN RU TE 2012 3 0161 2, POB 1-764, Bucharest 014700, Romania

关键词: Tonic ideals;    Matroids;    Graver basis;    Universal Grobner basis;    Hypergraphs;    Markov bases;    Resolutions;   

DOI  :  10.1016/j.jalgebra.2018.05.016

来源: Elsevier

【 摘 要 】

To any toric ideal I-A, encoded by an integer matrix A, we associate a matroid structure called the bouquet graph of A and introduce another tonic ideal called the bouquet ideal of A. We show how these objects capture the essential combinatorial and algebraic information about I-A . Passing from the toric ideal to its bouquet ideal via the graph theoretic properties of the bouquet graph allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call stable, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie tonic ideals whose various bases (e.g., minimal generating sets, Grobner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of tonic ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of tonic ideals whose Graver basis, the universal Grobner basis, any reduced Grobner basis and any minimal generating set coincide. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free