15 research outputs found
On moments of a polytope
We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a compact
polytope P in R^d is a rational function. Its denominator is the product of
linear forms dual to the vertices of P raised to the power equal to the degree
of the density function. Using this, we solve the inverse moment problem for
the set of, not necessarily convex, polytopes having a given set S of vertices.
Under a weak non-degeneracy assumption we also show that the uniform measure
supported on any such polytope is a linear combination of uniform measures
supported on simplices with vertices in S.Comment: 28 pages, 3 figure
Reconstructibility of matroid polytopes
We specify what is meant for a polytope to be reconstructible from its graph
or dual graph. And we introduce the problem of class reconstructibility, i.e.,
the face lattice of the polytope can be determined from the (dual) graph within
a given class. We provide examples of cubical polytopes that are not
reconstructible from their dual graphs. Furthermore, we show that matroid
(base) polytopes are not reconstructible from their graphs and not class
reconstructible from their dual graphs; our counterexamples include
hypersimplices. Additionally, we prove that matroid polytopes are class
reconstructible from their graphs, and we present a algorithm that
computes the vertices of a matroid polytope from its -vertex graph.
Moreover, our proof includes a characterisation of all matroids with isomorphic
basis exchange graphs.Comment: 22 pages, 5 figure
Signatures of partition functions and their complexity reduction through the KP II equation
A statistical amoeba arises from a real-valued partition function when the
positivity condition for pre-exponential terms is relaxed, and families of
signatures are taken into account. This notion lets us explore special types of
constraints when we focus on those signatures that preserve particular
properties. Specifically, we look at sums of determinantal type, and main
attention is paid to a distinguished class of soliton solutions of the
Kadomtsev-Petviashvili (KP) II equation. A characterization of the signatures
preserving the determinantal form, as well as the signatures compatible with
the KP II equation, is provided: both of them are reduced to choices of signs
for columns and rows of a coefficient matrix, and they satisfy the whole KP
hierarchy. Interpretations in term of information-theoretic properties,
geometric characteristics, and the relation with tropical limits are discussed.Comment: 42 pages, 11 figures. Section 7.1 has been added, the organization of
the paper has been change