817 research outputs found
Graph isomorphism and volumes of convex bodies
We show that a nontrivial graph isomorphism problem of two undirected graphs,
and more generally, the permutation similarity of two given
matrices, is equivalent to equalities of volumes of the induced three convex
bounded polytopes intersected with a given sequence of balls, centered at the
origin with radii , where is an increasing
sequence converging to . These polytopes are characterized by
inequalities in at most variables. The existence of fpras for computing
volumes of convex bodies gives rise to a semi-frpas of order at
most to find if given two undirected graphs are isomorphic.Comment: 9 page
Mixed volume and an extension of intersection theory of divisors
Let K(X) be the collection of all non-zero finite dimensional subspaces of
rational functions on an n-dimensional irreducible variety X. For any n-tuple
L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the
number of solutions in X of a system of equations f_1 = ... = f_n = 0 where
each f_i is a generic function from the space L_i. In counting the solutions,
we neglect the solutions x at which all the functions in some space L_i vanish
as well as the solutions at which at least one function from some subspace L_i
has a pole. The collection K(X) is a commutative semigroup with respect to a
natural multiplication. The intersection index [L_1,..., L_n] can be extended
to the Grothendieck group of K(X). This gives an extension of the intersection
theory of divisors. The extended theory is applicable even to non-complete
varieties. We show that this intersection index enjoys all the main properties
of the mixed volume of convex bodies. Our paper is inspired by the
Bernstein-Kushnirenko theorem from the Newton polytope theory.Comment: 31 pages. To appear in Moscow Mathematical Journa
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
Volume functions of linear series
The volume of a Cartier divisor is an asymptotic invariant, which measures
the rate of growth of sections of powers of the divisor. It extends to a
continuous, homogeneous, and log-concave function on the whole N\'eron--Severi
space, thus giving rise to a basic invariant of the underlying projective
variety. Analogously, one can also define the volume function of a possibly
non-complete multigraded linear series. In this paper we will address the
question of characterizing the class of functions arising on the one hand as
volume functions of multigraded linear series and on the other hand as volume
functions of projective varieties. In the multigraded setting, relying on the
work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that
any continuous, homogeneous, and log-concave function appears as the volume
function of a multigraded linear series. By contrast we show that there exists
countably many functions which arise as the volume functions of projective
varieties. We end the paper with an example, where the volume function of a
projective variety is given by a transcendental formula, emphasizing the
complicated nature of the volume in the classical case.Comment: 16 pages, minor revisio
LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies
We extend the classical LR characterization of chirotopes of finite planar
families of points to chirotopes of finite planar families of pairwise disjoint
convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a
chirotope of finite planar families of pairwise disjoint convex bodies if and
only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the
set of 3-subsets of J is a chirotope of finite planar families of pairwise
disjoint convex bodies. Our main tool is the polarity map, i.e., the map that
assigns to a convex body the set of lines missing its interior, from which we
derive the key notion of arrangements of double pseudolines, introduced for the
first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio
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