7,321 research outputs found
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
Finiteness theorems in stochastic integer programming
We study Graver test sets for families of linear multi-stage stochastic
integer programs with varying number of scenarios. We show that these test sets
can be decomposed into finitely many ``building blocks'', independent of the
number of scenarios, and we give an effective procedure to compute these
building blocks. The paper includes an introduction to Nash-Williams' theory of
better-quasi-orderings, which is used to show termination of our algorithm. We
also apply this theory to finiteness results for Hilbert functions.Comment: 36 p
Single-shot fault-tolerant quantum error correction
Conventional quantum error correcting codes require multiple rounds of
measurements to detect errors with enough confidence in fault-tolerant
scenarios. Here I show that for suitable topological codes a single round of
local measurements is enough. This feature is generic and is related to
self-correction and confinement phenomena in the corresponding quantum
Hamiltonian model. 3D gauge color codes exhibit this single-shot feature, which
applies also to initialization and gauge-fixing. Assuming the time for
efficient classical computations negligible, this yields a topological
fault-tolerant quantum computing scheme where all elementary logical operations
can be performed in constant time.Comment: Typos corrected after publication in journal, 26 pages, 4 figure
Geometric and homological finiteness in free abelian covers
We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics
and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201
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