276 research outputs found
Tetrahedral curves via graphs and Alexander duality
A tetrahedral curve is a (usually nonreduced) curve in P^3 defined by an
unmixed, height two ideal generated by monomials. We characterize when these
curves are arithmetically Cohen-Macaulay by associating a graph to each curve
and, using results from combinatorial commutative algebra and Alexander
duality, relating the structure of the complementary graph to the
Cohen-Macaulay property.Comment: 15 pages; minor revisions to v. 1 to improve clarity; to appear in
JPA
Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
We revisit a well-known family of polynomial ideals encoding the problem of
graph--colorability. Our paper describes how the inherent combinatorial
structure of the ideals implies several interesting algebraic properties.
Specifically, we provide lower bounds on the difficulty of computing Gr\"obner
bases and Nullstellensatz certificates for the coloring ideals of general
graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis
for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page
Edge ideals: algebraic and combinatorial properties
Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on
the algebraic and combinatorial properties of R/I(C) and C, respectively. We
give a criterion to estimate the regularity of R/I(C) and apply this criterion
to give new proofs of some formulas for the regularity. If C is a clutter and
R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity
of the ideal of vertex covers of C and give a formula for the projective
dimension of R/I(C). We also examine the associated primes of powers of edge
ideals, and show that for a graph with a leaf, these sets form an ascending
chain
Root polytopes, tropical types, and toric edge ideals
We consider arrangements of tropical hyperplanes where the apices of the
hyperplanes are taken to infinity in certain directions. Such an arrangement
defines a decomposition of Euclidean space where a cell is determined by its
`type' data, analogous to the covectors of an oriented matroid. By work of
Develin-Sturmfels and Fink-Rinc\'{o}n, these `tropical complexes' are dual to
(regular) subdivisions of root polytopes, which in turn are in bijection with
mixed subdivisions of certain generalized permutohedra. Extending previous work
with Joswig-Sanyal, we show how a natural monomial labeling of these complexes
describes polynomial relations (syzygies) among `type ideals' which arise
naturally from the combinatorial data of the arrangement. In particular, we
show that the cotype ideal is Alexander dual to a corresponding initial ideal
of the lattice ideal of the underlying root polytope. This leads to novel ways
of studying algebraic properties of various monomial and toric ideals, as well
as relating them to combinatorial and geometric properties. In particular, our
methods of studying the dimension of the tropical complex leads to new formulas
for homological invariants of toric edge ideals of bipartite graphs, which have
been extensively studied in the commutative algebra community.Comment: 45 page
Cohen-Macaulay Weighted Oriented Chordal and Simplicial Graphs
Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal
graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex)
weighted oriented chordal and simplicial graphs, a more general class of
monomial ideals. In particular, we show that the Cohen-Macaulay property of
these ideals is equivalent to the unmixed one and hence, independent of the
underlying field.Comment: 7 pages, 1 figur
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.Comment: 40 pages, 5 figure
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