scholarly journals Toric Ideals Generated by Circuits

2012 ◽  
Vol 19 (04) ◽  
pp. 665-672 ◽  
Author(s):  
José Martínez-Bernal ◽  
Rafael H. Villarreal
Keyword(s):  
Linear Combination ◽  
Free Term ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Normal Configuration

Let [Formula: see text] be the toric ideal of a homogeneous normal configuration [Formula: see text]. We prove that [Formula: see text] is generated by circuits if and only if each unbalanced circuit of [Formula: see text] has a connector which is a linear combination of circuits with a square-free term. In particular, if each circuit of [Formula: see text] with non-square-free terms is balanced, then [Formula: see text] is generated by circuits. As a consequence, we prove that the toric ideal of a normal edge subring of a multigraph is generated by circuits with a square-free term.

scholarly journals Graphs and complete intersection toric ideals

2015 ◽  
Vol 14 (09) ◽  
pp. 1540011 ◽  
Author(s):  
I. Bermejo ◽  
I. García-Marco ◽  
E. Reyes
Keyword(s):  
Complete Intersection ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Toric Ideals ◽  
Induced Subgraphs ◽  
Toric Ideal ◽  
Vertex Set ◽  
Combinatorial Point ◽  
Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

scholarly journals Betti numbers of toric ideals of graphs: A case study

2019 ◽  
Vol 18 (12) ◽  
pp. 1950226
Author(s):  
Federico Galetto ◽  
Johannes Hofscheier ◽  
Graham Keiper ◽  
Craig Kohne ◽  
Adam Van Tuyl ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Hilbert Series ◽  
Complete Bipartite Graph ◽  
Betti Numbers ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Initial Ideal ◽  
Graded Betti Numbers ◽  
Linear Quotients

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and [Formula: see text]-vector for all the toric ideals of graphs in this family.

scholarly journals Toric Rings and Ideals of Stable Set Polytopes

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 613
Author(s):  
Kazunori Matsuda ◽  
Hidefumi Ohsugi ◽  
Kazuki Shibata
Keyword(s):  
Gröbner Bases ◽  
Infinite Family ◽  
Grobner Bases ◽  
Stable Set ◽  
Stability Number ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Stable Set Polytope ◽  
Toric Rings

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases.

scholarly journals Toric varieties and Gröbner bases: the complete $$\mathbb {Q}$$-factorial case

2020 ◽  
Vol 31 (5-6) ◽  
pp. 461-482
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Toric Varieties ◽  
Toric Ideals ◽  
Projective Embedding ◽  
Complete Case ◽  
Open Problems ◽  
Toric Ideal ◽  
Definition Of ◽  
Nef Cone ◽  
Special Fibre

Abstract We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by V and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels’ arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to V allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by V. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $$\mathbb {Q}$$ Q -factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

On the binomial arithmetical rank of toric ideals

2014 ◽  
Vol 13 (07) ◽  
pp. 1450030 ◽  
Author(s):  
Anargyros Katsabekis
Keyword(s):  
Sufficient Conditions ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Number Of Generators ◽  
Arithmetical Rank

In this paper, we investigate sufficient conditions which ensure that the minimal number of generators of a toric ideal IA is equal to its binomial arithmetical rank. We study particularly the case where IA is the toric ideal of a graph.

scholarly journals On the toric ideals of matroids of a fixed rank

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Michał Lasoń
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Fixed Rank

AbstractIn 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 (Lasoń and Michałek in Adv Math 259:1–12, 2014) 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.

scholarly journals Strongly Robust Toric Ideals in Codimension 2

2019 ◽  
Vol 10 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Seth Sullivant
Keyword(s):  
Positive Answer ◽  
Homogeneous Ideal ◽  
Toric Ideals ◽  
Grobner Basis ◽  
Toric Ideal ◽  
Generating Set ◽  
Graver Basis ◽  
Minimal Generating Set ◽  
Gale Diagrams ◽  
Universal Gröbner Basis

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set.  For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials.  We characterize the codimension 2  strongly robust toric ideals by their Gale diagrams.  This give a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

scholarly journals Toric Ideals of Finite Graphs and Adjacent 2-Minors

2014 ◽  
Vol 114 (2) ◽  
pp. 185 ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Takayuki Hibi
Keyword(s):  
Finite Graph ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Finite Graphs

We study the problem when an ideal generated by adjacent $2$-minors is the toric ideal of a finite graph.

INDISPENSABLE BINOMIALS OF FINITE GRAPHS

2005 ◽  
Vol 04 (04) ◽  
pp. 421-434 ◽  
Author(s):  
HIDEFUMI OHSUGI ◽  
TAKAYUKI HIBI
Keyword(s):  
Complete Graph ◽  
Finite Graph ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Finite Graphs ◽  
Cycle Condition ◽  
Odd Cycle ◽  
Complementary Graph

A binomial f belonging to a toric ideal I is indispensable if, for any system [Formula: see text] of binomial generators of I, either f or -f belongs to [Formula: see text]. In the present paper, we study indispensable binomials of the toric ideals IG arising from a finite graph G. First, we show that the toric ideal IG arising from a finite graph G whose complementary graph is weakly chordal is generated by the indispensable binomials if and only if no complete graph of order ≥4 is a subgraph of G. Second, we completely classify indispensable binomials of the toric ideal IG arising from a finite graph G satisfying the odd cycle condition. Finally, the existence of indispensable binomials of IG will be discussed.

scholarly journals Toric ideals of series and parallel connections of matroids

2016 ◽  
Vol 15 (06) ◽  
pp. 1650106 ◽  
Author(s):  
Kazuki Shibata
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Parallel Connections

In 1980, White conjectured that the toric ideal associated to a matroid is generated by binomials corresponding to a symmetric exchange. In this paper, we prove that classes of matroids for which the toric ideal is generated by quadrics and that has quadratic Gröbner bases, are closed under series and parallel extensions, series and parallel connections, and 2-sums.

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