185 research outputs found
Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
A general issue in computational optimization is to develop combinatorial
algorithms for semidefinite programming. We address this issue when the base
field is nonarchimedean. We provide a solution for a class of semidefinite
feasibility problems given by generic matrices. Our approach is based on
tropical geometry. It relies on tropical spectrahedra, which are defined as the
images by the valuation of nonarchimedean spectrahedra. We establish a
correspondence between generic tropical spectrahedra and zero-sum stochastic
games with perfect information. The latter have been well studied in
algorithmic game theory. This allows us to solve nonarchimedean semidefinite
feasibility problems using algorithms for stochastic games. These algorithms
are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions +
benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign
patterns + some results have been replaced by references to the companion
work arXiv:1610.0674
Affine semigroups having a unique Betti element
We characterize affine semigroups having one Betti element and we compute
some relevant non-unique factorization invariants for these semigroups. As an
example, we particularize our description to numerical semigroups.Comment: 8 pages, 1 figure. To appear in Journal of Algebra and its
Application
Quantum Correlations in the Minimal Scenario
In the minimal scenario of quantum correlations, two parties can choose from
two observables with two possible outcomes each. Probabilities are specified by
four marginals and four correlations. The resulting four-dimensional convex
body of correlations, denoted , is fundamental for quantum
information theory. It is here studied through the lens of convex algebraic
geometry. We review and systematize what is known and add many details,
visualizations, and complete proofs. A new result is that is
isomorphic to its polar dual. The boundary of consists of
three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds
of exposed extreme points. These share all basic properties with the usual
maximally CHSH-violating correlations. These patches are separated by cubic
surfaces of non-exposed extreme points. We provide a trigonometric
parametrization of all extreme points, along with their exposing Tsirelson
inequalities and quantum models. All non-classical extreme points (exposed or
not) are self-testing, i.e., realized by an essentially unique quantum model.
Two principles, which are specific to the minimal scenario, allow a quick and
complete overview: The first is the pushout transformation, the application of
the sine function to each coordinate. This transforms the classical polytope
exactly into the correlation body , also identifying the boundary
structures. The second principle, self-duality, reveals the polar dual, i.e.,
the set of all Tsirelson inequalities satisfied by all quantum correlations.
The convex body includes the classical correlations, a cross
polytope, and is contained in the no-signaling body, a 4-cube. These polytopes
are dual to each other, and the linear transformation realizing this duality
also identifies with its dual.Comment: We also discuss the sets of correlations achieved with fixed Hilbert
space dimension, fixed state or fixed observable
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Separability problem for multipartite states of rank at most four
One of the most important problems in quantum information is the separability
problem, which asks whether a given quantum state is separable. We investigate
multipartite states of rank at most four which are PPT (i.e., all their partial
transposes are positive semidefinite). We show that any PPT state of rank two
or three is separable and has length at most four. For separable states of rank
four, we show that they have length at most six. It is six only for some
qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of
rank four is necessarily supported on a 3x3 or a 2x2x2 subsystem. We obtain a
very simple criterion for the separability problem of the PPT states of rank at
most four: such a state is entangled if and only if its range contains no
product vectors. This criterion can be easily applied since a four-dimensional
subspace in the 3x3 or 2x2x2 system contains a product vector if and only if
its Pluecker coordinates satisfy a homogeneous polynomial equation (the Chow
form of the corresponding Segre variety). We have computed an explicit
determinantal expression for the Chow form in the former case, while such
expression was already known in the latter case.Comment: 19 page
Affine convex body semigroups
In this paper we present a new kind of semigroups called convex body
semigroups which are generated by convex bodies of R^k. They generalize to
arbitrary dimension the concept of proportionally modular numerical semigroup
of [7]. Several properties of these semigroups are proven. Affine convex body
semigroups obtained from circles and polygons of R^2 are characterized. The
algorithms for computing minimal system of generators of these semigroups are
given. We provide the implementation of some of them
Factorization invariants in half-factorial affine semigroups
Let be the monoid generated by We introduce the
homogeneous catenary degree of as the smallest with the following property: for each and any two factorizations of
, there exists factorizations of such that, for every where is the
usual distance between factorizations, and the length of is less than or equal to
We prove that the homogeneous catenary degree of
improves the monotone catenary degree as upper bound for the ordinary catenary
degree, and we show that it can be effectively computed. We also prove that for
half-factorial monoids, the tame degree and the -primality coincide,
and that all possible catenary degrees of the elements of an affine semigroup
of this kind occur as the catenary degree of one of its Betti elements.Comment: 8 pages, 1 figur
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
Design and analysis of fractional factorial experiments from the viewpoint of computational algebraic statistics
We give an expository review of applications of computational algebraic
statistics to design and analysis of fractional factorial experiments based on
our recent works. For the purpose of design, the techniques of Gr\"obner bases
and indicator functions allow us to treat fractional factorial designs without
distinction between regular designs and non-regular designs. For the purpose of
analysis of data from fractional factorial designs, the techniques of Markov
bases allow us to handle discrete observations. Thus the approach of
computational algebraic statistics greatly enlarges the scope of fractional
factorial designs.Comment: 16 page
A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope
We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric
- …