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 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.

Binomial fibers and indispensable binomials

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.

Markov complexity of monomial curves

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 $l$.

Markov bases and generalized Lawrence liftings

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 universal 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.

Minimal Generating Sets of Lattice Ideals

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.