82 research outputs found
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
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
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
Combinatorial symbolic powers
Symbolic powers are studied in the combinatorial context of monomial ideals.
When the ideals are generated by quadratic squarefree monomials, the generators
of the symbolic powers are obstructions to vertex covering in the associated
graph and its blowups. As a result, perfect graphs play an important role in
the theory, dual to the role played by perfect graphs in the theory of secants
of monomial ideals. We use Gr\"obner degenerations as a tool to reduce
questions about symbolic powers of arbitrary ideals to the monomial case. Among
the applications are a new, unified approach to the Gr\"obner bases of symbolic
powers of determinantal and Pfaffian ideals.Comment: 29 pages, 3 figures, Positive characteristic results incorporated
into main body of pape
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