19,705 research outputs found
Implicitization of surfaces via geometric tropicalization
In this paper we further develop the theory of geometric tropicalization due
to Hacking, Keel and Tevelev and we describe tropical methods for
implicitization of surfaces. More precisely, we enrich this theory with a
combinatorial formula for tropical multiplicities of regular points in
arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev
regarding sufficient combinatorial conditions to compute tropical varieties via
geometric tropicalization. Using these two results, we extend previous work of
Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and
we provide methods for approaching the non-generic cases.Comment: 20 pages, 6 figures. Mayor reorganization and exposition improved.
The results on geometric tropicalization have been extended to any dimension.
In particular, Conjecture 2.8 is now Theorem 2.
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
Isogeny graphs with maximal real multiplication
An isogeny graph is a graph whose vertices are principally polarized abelian
varieties and whose edges are isogenies between these varieties. In his thesis,
Kohel described the structure of isogeny graphs for elliptic curves and showed
that one may compute the endomorphism ring of an elliptic curve defined over a
finite field by using a depth first search algorithm in the graph. In dimension
2, the structure of isogeny graphs is less understood and existing algorithms
for computing endomorphism rings are very expensive. Our setting considers
genus 2 jacobians with complex multiplication, with the assumptions that the
real multiplication subring is maximal and has class number one. We fully
describe the isogeny graphs in that case. Over finite fields, we derive a depth
first search algorithm for computing endomorphism rings locally at prime
numbers, if the real multiplication is maximal. To the best of our knowledge,
this is the first DFS-based algorithm in genus 2
Formal Desingularization of Surfaces - The Jung Method Revisited -
In this paper we propose the concept of formal desingularizations as a
substitute for the resolution of algebraic varieties. Though a usual resolution
of algebraic varieties provides more information on the structure of
singularities there is evidence that the weaker concept is enough for many
computational purposes. We give a detailed study of the Jung method and show
how it facilitates an efficient computation of formal desingularizations for
projective surfaces over a field of characteristic zero, not necessarily
algebraically closed. The paper includes a generalization of Duval's Theorem on
rational Puiseux parametrizations to the multivariate case and a detailed
description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
Loop Quantum Gravity defines the quantum states of space geometry as spin
networks and describes their evolution in time. We reformulate spin networks in
terms of harmonic oscillators and show how the holographic degrees of freedom
of the theory are described as matrix models. This allow us to make a link with
non-commutative geometry and to look at the issue of the semi-classical limit
of LQG from a new perspective. This work is thought as part of a bigger project
of describing quantum geometry in quantum information terms.Comment: 16 pages, revtex, 3 figure
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