168 research outputs found
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate
quadratic equations in n variables have 2 n solutions over algebraically closed
fields. When only a small subset M of monomials appear in the equations
(fewnomial systems), the number of solutions may decrease dramatically. We
focus in this work on subsets of quadratic monomials M such that generic
systems with support M do not admit any solution at all. For these systems,
Hilbert's Nullstellensatz ensures the existence of algebraic certificates of
inconsistency. However, up to our knowledge all known bounds on the sizes of
such certificates -including those which take into account the Newton polytopes
of the polynomials- are exponential in n. Our main results show that if the
inequality 2|M| -- 2n \sqrt 1 + 8{\nu} -- 1 holds for a quadratic
fewnomial system -- where {\nu} is the matching number of a graph associated
with M, and |M| is the cardinality of M -- then there exists generically a
certificate of inconsistency of linear size (measured as the number of
coefficients in the ground field K). Moreover this certificate can be computed
within a polynomial number of arithmetic operations. Next, we evaluate how
often this inequality holds, and we give evidence that the probability that the
inequality is satisfied depends strongly on the number of squares. More
precisely, we show that if M is picked uniformly at random among the subsets of
n + k + 1 quadratic monomials containing at least (n 1/2+)
squares, then the probability that the inequality holds tends to 1 as n grows.
Interestingly, this phenomenon is related with the matching number of random
graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results
showing that certificates in inconsistency can be computed for systems with
more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201
On the dimension of subspaces with bounded Schmidt rank
We consider the question of how large a subspace of a given bipartite quantum
system can be when the subspace contains only highly entangled states. This is
motivated in part by results of Hayden et al., which show that in large d x
d--dimensional systems there exist random subspaces of dimension almost d^2,
all of whose states have entropy of entanglement at least log d - O(1). It is
also related to results due to Parthasarathy on the dimension of completely
entangled subspaces, which have connections with the construction of
unextendible product bases. Here we take as entanglement measure the Schmidt
rank, and determine, for every pair of local dimensions dA and dB, and every r,
the largest dimension of a subspace consisting only of entangled states of
Schmidt rank r or larger. This exact answer is a significant improvement on the
best bounds that can be obtained using random subspace techniques. We also
determine the converse: the largest dimension of a subspace with an upper bound
on the Schmidt rank. Finally, we discuss the question of subspaces containing
only states with Schmidt equal to r.Comment: 4 pages, REVTeX4 forma
One-dimensional polynomial maps, periodic points and multipliers
We discuss tangent maps related to the multipliers of periodic points of a
typical one-dimensional polynomial map.Comment: We correct inaccuracies (wrong signs) in the computation of gradients
of multiplier functions (Sect. 2). The statements and proofs of main results
remain unchange
Punctual Hilbert Schemes and Certified Approximate Singularities
In this paper we provide a new method to certify that a nearby polynomial
system has a singular isolated root with a prescribed multiplicity structure.
More precisely, given a polynomial system f , we present a Newton iteration on an extended deflated system
that locally converges, under regularity conditions, to a small deformation of
such that this deformed system has an exact singular root. The iteration
simultaneously converges to the coordinates of the singular root and the
coefficients of the so called inverse system that describes the multiplicity
structure at the root. We use -theory test to certify the quadratic
convergence, and togive bounds on the size of the deformation and on the
approximation error. The approach relies on an analysis of the punctual Hilbert
scheme, for which we provide a new description. We show in particular that some
of its strata can be rationally parametrized and exploit these parametrizations
in the certification. We show in numerical experimentation how the approximate
inverse system can be computed as a starting point of the Newton iterations and
the fast numerical convergence to the singular root with its multiplicity
structure, certified by our criteria.Comment: International Symposium on Symbolic and Algebraic Computation, Jul
2020, Kalamata, Franc
On the Symmetries of Integrability
We show that the Yang-Baxter equations for two dimensional models admit as a
group of symmetry the infinite discrete group . The existence of
this symmetry explains the presence of a spectral parameter in the solutions of
the equations. We show that similarly, for three-dimensional vertex models and
the associated tetrahedron equations, there also exists an infinite discrete
group of symmetry. Although generalizing naturally the previous one, it is a
much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to
resolve the Yang-Baxter equations and their higher-dimensional generalizations
and initiate the study of three-dimensional vertex models. These symmetries are
naturally represented as birational projective transformations. They may
preserve non trivial algebraic varieties, and lead to proper parametrizations
of the models, be they integrable or not. We mention the relation existing
between spin models and the Bose-Messner algebras of algebraic combinatorics.
Our results also yield the generalization of the condition so often
mentioned in the theory of quantum groups, when no parameter is available.Comment: 23 page
The Topology of Parabolic Character Varieties of Free Groups
Let G be a complex affine algebraic reductive group, and let K be a maximal
compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or
equal to 0, let X (respectively, Y) be the space of equivalence classes of
representations of the free group of m+n generators in G (respectively, K) such
that for each i between 1 and m, the image of the i-th free generator is
conjugate to h_i. These spaces are parabolic analogues of character varieties
of free groups. We prove that Y is a strong deformation retraction of X. In
particular, X and Y are homotopy equivalent. We also describe explicit examples
relating X to relative character varieties.Comment: 16 pages, version 2 includes minor revisions and some modified
proofs, accepted for publication in Geometriae Dedicat
Total Degree Formula for the Generic Offset to a Parametric Surface
We provide a resultant-based formula for the total degree w.r.t. the spatial
variables of the generic offset to a parametric surface. The parametrization of
the surface is not assumed to be proper.Comment: Preprint of an article to be published at the International Journal
of Algebra and Computation, World Scientific Publishing,
DOI:10.1142/S021819671100680
Nonlinear analysis of spacecraft thermal models
We study the differential equations of lumped-parameter models of spacecraft
thermal control. Firstly, we consider a satellite model consisting of two
isothermal parts (nodes): an outer part that absorbs heat from the environment
as radiation of various types and radiates heat as a black-body, and an inner
part that just dissipates heat at a constant rate. The resulting system of two
nonlinear ordinary differential equations for the satellite's temperatures is
analyzed with various methods, which prove that the temperatures approach a
steady state if the heat input is constant, whereas they approach a limit cycle
if it varies periodically. Secondly, we generalize those methods to study a
many-node thermal model of a spacecraft: this model also has a stable steady
state under constant heat inputs that becomes a limit cycle if the inputs vary
periodically. Finally, we propose new numerical analyses of spacecraft thermal
models based on our results, to complement the analyses normally carried out
with commercial software packages.Comment: 29 pages, 4 figure
Universal Calabi-Yau Algebra: Towards an Unification of Complex Geometry
We present a universal normal algebra suitable for constructing and
classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach
includes natural extensions of reflexive weight vectors to higher dimensions,
related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also
includes a `dual' construction based on the Diophantine decomposition of
invariant monomials, which provides explicit recurrence formulae for the
numbers of Calabi-Yau spaces in arbitrary dimensions with Weierstrass, K3,
etc., fibrations. Our approach also yields simple algebraic relations between
chains of Calabi-Yau spaces in different dimensions, and concrete
visualizations of their singularities related to Cartan-Lie algebras. This
Universal Calabi-Yau Algebra is a powerful tool for decyphering the Calabi-Yau
genome in all dimensions.Comment: 81 pages LaTeX, 8 eps figure
Algebraic varieties in Birkhoff strata of the Grassmannian Gr: Harrison cohomology and integrable systems
Local properties of families of algebraic subsets in Birkhoff strata
of Gr containing hyperelliptic curves of genus are
studied. It is shown that the tangent spaces for are isomorphic to
linear spaces of 2-coboundaries. Particular subsets in are described by
the intergrable dispersionless coupled KdV systems of hydrodynamical type
defining a special class of 2-cocycles and 2-coboundaries in . It is
demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and
gradient catastrophes for associated integrable systems are interrelated.Comment: 28 pages, no figures. Generally improved version, in particular the
Discussion section. Added references. Corrected typo
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