185 research outputs found
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
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
Equality of Graver bases and universal Gr\"obner bases of colored partition identities
Associated to any vector configuration A is a toric ideal encoded by vectors
in the kernel of A. Each toric ideal has two special generating sets: the
universal Gr\"obner basis and the Graver basis. While the former is generally a
proper subset of the latter, there are cases for which the two sets coincide.
The most prominent examples among them are toric ideals of unimodular matrices.
Equality of universal Gr\"obner basis and Graver basis is a combinatorial
property of the toric ideal (or, of the defining matrix), providing interesting
information about ideals of higher Lawrence liftings of a matrix. Nonetheless,
a general classification of all matrices for which both sets agree is far from
known. We contribute to this task by identifying all cases with equality within
two families of matrices; namely, those defining rational normal scrolls and
those encoding homogeneous primitive colored partition identities.Comment: minor revision; references added; introduction expanded
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