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 free resolution. On the other end of the spectrum lie toric 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 toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Grobner basis, any reduced Grobner basis and any minimal generating set coincide.
Let $I$ be an arbitrary ideal generated by binomials. We show that certain equivalence classes of fibers are associated to any minimal binomial generating set of $I$. We provide a simple and efficient algorithm to compute the indispensable binomials
of a binomial ideal from a given generating set of binomials and an algorithm to detect whether a binomial ideal is generated by indispensable binomials.
Let $mathcal{A}={{bf a}_1,ldots,{bf a}_n}subsetBbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{mathcal{A}}$. We show that the Markov complexity of $mathcal{A}={n_1,n_2,n_3}$ is equal to two if $I_
{mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $rgeq 2$ there is a unique minimal Markov basis of $mathcal{A}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $mathcal{A}$ is greater than $l$.
Let $I_A$ be a toric ideal. We prove that the degrees of the elements of the Graver basis of $I_A$ are not polynomially bounded by the true degrees of the circuits of $I_A$.
Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the uni
versal Gr{ o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $Ainmathcal{M}_{mtimes n}(Bbb{Z})$ and $Binmathcal{M}_{ptimes n}(Bbb{Z})$ and show that in cases of interest the {em complexity} of any two Markov bases is the same.
Let $Lsubset mathbb{Z}^n$ be a lattice and $I_L=langle x^{bf u}-x^{bf v}: {bf u}-{bf v}in Lrangle$ be the corresponding lattice ideal in $Bbbk[x_1,ldots, x_n]$, where $Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I
_L$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $I_L$. As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We pro
ve that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.
The universal Gr{o}bner basis of $I$, is a Gr{o}bner basis for $I$ with respect to all term orders simultaneously. Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the elements of the universal Gr{o}bner basis o
f the toric ideal $I_G$. We provide a bound for the degree of the binomials in the universal Gr{o}bner basis of the toric ideal of a graph. Finally we give a family of examples of circuits for which their true degrees are less than the degrees of some elements of the Graver basis.
Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.
Let $k$ be a field, $ mathcal{L}subset mathbb{Z}^n$ be a lattice such that $Lcap mathbb{N}^n={{bf 0}}$, and $I_Lsubset Bbbk[x_1,..., x_n]$ the corresponding lattice ideal. We present the generalized Scarf complex of $I_L$ and show that it is indispen
sable in the sense that it is contained in every minimal free resolution of $R/I_L$.
