40 research outputs found
Indispensable monomials of toric ideals and Markov bases
Extending the notion of indispensable binomials of a toric ideal, we define
indispensable monomials of a toric ideal and establish some of their
properties. They are useful for searching indispensable binomials of a toric
ideal and for proving the existence or non-existence of a unique minimal system
of binomials generators of a toric ideal. Some examples of indispensable
monomials from statistical models for contingency tables are given.Comment: 20 pages, 5 figure
Indispensable binomials in semigroup ideals
In this paper, we deal with the problem of uniqueness of minimal system of
binomial generators of a semigroup ideal. Concretely, we give different
necessary and/or sufficient conditions for uniqueness of such minimal system of
generators. These conditions come from the study and combinatorial description
of the so-called indispensable binomials in the semigroup ideal.Comment: 11 pages. This paper was initially presented at the II Iberian
Mathematical Meeting (http://imm2.unex.es). To appear in the Proc. Amer.
Math. So
Combinatorial degree bound for toric ideals of hypergraphs
Associated to any hypergraph is a toric ideal encoding the algebraic
relations among its edges. We study these ideals and the combinatorics of their
minimal generators, and derive general degree bounds for both uniform and
non-uniform hypergraphs in terms of balanced hypergraph bicolorings,
separators, and splitting sets. In turn, this provides complexity bounds for
algebraic statistical models associated to hypergraphs. As two main
applications, we recover a well-known complexity result for Markov bases of
arbitrary 3-way tables, and we show that the defining ideal of the tangential
variety is generated by quadratics and cubics in cumulant coordinates.Comment: Revised, improved, reorganized. We recommend viewing figures in colo
Minimal systems of binomial generators and the indispensable complex of a toric ideal
Let be a vector
configuration and its corresponding toric ideal.
The paper consists of two parts. In the first part we completely determine the
number of different minimal systems of binomial generators of . We also
prove that generic toric ideals are generated by indispensable binomials. In
the second part we associate to a simplicial complex \Delta _{\ind(A)}.
We show that the vertices of \Delta_{\ind(A)} correspond to the indispensable
monomials of the toric ideal , while one dimensional facets of
\Delta_{\ind(A)} with minimal binomial -degree correspond to the
indispensable binomials of
The three-state toric homogeneous Markov chain model has Markov degree two
We prove that the three-state toric homogenous Markov chain model has Markov
degree two. In algebraic terminology this means, that a certain class of toric
ideals are generated by quadratic binomials. This was conjectured by Haws,
Martin del Campo, Takemura and Yoshida, who proved that they are generated by
binomials of degree six or less.Comment: Updated language and notation. 13page
Two way subtable sum problems and quadratic Groebner bases
Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable
sum problems and shows that these toric ideals are generated by quadratic
binomials if and only if the subtables are either diagonal or triangular. In
the present paper, we show that if the subtables are either diagonal or
triangular, then their toric ideals possess quadratic Groebner bases.Comment: 3 page