Toric Ideals Generated by Circuits
نویسندگان
چکیده
Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a “connector” which is a linear combination of circuits with a square-free term. In particular if each circuit of IA with non-square-free terms is balanced, then IA 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.
منابع مشابه
A pr 2 00 9 TORIC IDEALS GENERATED BY CIRCUITS
Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a connector which is a linear combination of circuits with a square-free term. In particular, if each circuit of IA with non-square-free terms is balanced, then IA is generated by circuits. As a consequence we prove that the toric ide...
متن کاملMatchings in simplicial complexes, circuits and toric varieties
Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties. © 2006 Elsevier Inc. All rights reserved.
متن کاملRing graphs and complete intersection toric ideals
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideal...
متن کاملQuadratic Gröbner bases for smooth 3 × 3 transportation polytopes
The toric ideals of 3 × 3 transportation polytopes Trc are quadratically generated. The only exception is the Birkhoff polytope B3. If Trc is not a multiple of B3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3 × 3 transportation polytopes.
متن کاملMorphisms and Order Ideals of Toric Posets
Abstract: Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of...
متن کامل