Gröbner bases of toric ideals associated with matroids

On the toric ideals of matroids of a fixed rank

In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result arXiv:1302.5236 confirming White's conjecture `up to saturation'. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank.

On the toric ideal of a matroid

Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.

Gorenstein graphic matroids from multigraphs

A matroid is Gorenstein if its toric variety is. Hibi, Lasoń, Matsuda, Michałek, and Vodička provided a full graph-theoretic classification of Gorenstein matroids associated to simple graphs. We extend this classification to multigraphs.

存在无二次 Gröbner 基的拟阵环面理想

There are matroid toric ideals without quadratic Gröbner bases

本文证明若拟阵包含Fano平面或其对偶作为子式，则其环面理想无二次Gröbner基，通过建立与正则单模旗三角剖分的联系并利用多面体重新表述解决Herzog和Hibi的著名问题。

matroid 的秩-空环

The rank-nullity ring of a matroid

本文提出 matroid 的秩-空环概念，并在均匀 matroid 情况下计算其 Hilbert 函数和 Gröbner 基。

White猜想的Gröbner版本是错误的

The Gröbner Version of White's Conjecture is False

通过证明Fano拟阵多面体的环面理想没有二次Gröbner基，否定了拟阵论中White猜想的强版本。

The slack realization space of a matroid

We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is inspired by a similar model for the slack realization space of a polytope. We show how to use these ideas to certify non-realizability of matroids, and describe an explicit relationship to the standard Grassmann-Plücker realization space model. We also exhibit a way of detecting projectively unique matroids via their slack ideals by introducing a toric idea…

On the strongly robustness property of toric ideals

To every toric ideal one can associate an oriented matroid structure, consisting of a graph and another toric ideal, called bouquet ideal. The connected components of this graph are called bouquets. Bouquets are of three types; free, mixed and non mixed. We prove that the cardinality of the following sets - the set of indispensable elements, minimal Markov bases, the Universal Markov basis and the Universal Gröbner basis of a toric ideal - depends only on the type of the bouquets and the bouquet ideal. These results enable us to introduce the strongly robustness simplicial complex and show tha…

Bouquet algebra of toric ideals

To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {\em the bouquet graph} of $A$ and introduce another toric ideal called {\em 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 reveals a structure that allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call {\em stable}, for which bouquets capture the complexity of various generating sets as well as the minimal fr…

Varieties of Tropical Ideals are Balanced

Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of var…

根多面体、热带类型与环面边理想

Root polytopes, tropical types, and toric edge ideals

本文利用热带超平面构形、根多面体细分与类型理想之间的对偶关系，证明余类型理想的 Alexander 对偶性质，并导出二部图环面边理想同调不变量的新公式。

Products of polymatroids with the strong exchange property

It was conjectured by White in 1980 that the toric ring associated to a matroid is defined by symmetric exchange relations. This conjecture was extended to discrete polymatroids by Herzog and Hibi, and they prove that the conjecture holds for polymatroids with the so called strong exchange property. In this paper we generalize their result to polymatroids that are products of polymatroids with the strong exchange property. This also extends a result by Conca on transversal polymatroids.