Quadratic Grobner Basis Research Articles

Toric geometry of the Cavender-Farris-Neyman model with a molecular clock

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial “cluster” structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic Gröbner basis with squarefree initial terms. Finally, we show that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself. These results are analogous to classic results for the Cavender-Farris-Neyman model without a molecular clock. However, new techniques are required because the molecular clock assumption destroys the toric fiber product structure that governs group-based models without the molecular clock.

The Rees algebra of a two-Borel ideal is Koszul

Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Grobner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

Strongly Koszul Edge Rings

We classify the finite connected simple graphs whose edge rings are strongly Koszul. From the classification, it follows that if the edge ring is strongly Koszul, then its toric ideal possesses a quadratic Grobner basis.

Powers of lexsegment ideals with linear resolution

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of lexsegment ideals, the corresponding Rees algebra has a quadratic Grobner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

Ideals of fiber type and polymatroids

In the first half of this paper, we complement the theory on discrete polymatroids. More precisely, (i) we prove that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the other hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Grobner basis. For this purpose we introduce the concept of ``$l$-exchange property'' and show that a Grobner basis of the defining ideal of the Rees ring of an ideal $I$ can be determined and that $I$ is of fiber type if $I$ satisfies the $l$-exchange property. Ideals satisfying the $l$-exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.

Distributive Lattices, Bipartite Graphs and Alexander Duality

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Grobner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be constructed explicitly and a combinatorial formula to compute the Betti numbers of HP will be presented. Third, by using the fact that the Alexander dual of the simplicial complex ? whose Stanley---Reisner ideal coincides with HP is Cohen---Macaulay, all the Cohen---Macaulay bipartite graphs will be classified.

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Grobner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Grobner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.