1,937 research outputs found
Sparse Gr\"obner Bases: the Unmixed Case
Toric (or sparse) elimination theory is a framework developped during the
last decades to exploit monomial structures in systems of Laurent polynomials.
Roughly speaking, this amounts to computing in a \emph{semigroup algebra},
\emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to
solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases},
an analog of classical Gr\"obner bases for semigroup algebras, and we propose
sparse variants of the and FGLM algorithms to compute them. Our prototype
"proof-of-concept" implementation shows large speed-ups (more than 100 for some
examples) compared to optimized (classical) Gr\"obner bases software. Moreover,
in the case where the generating subset of monomials corresponds to the points
with integer coordinates in a normal lattice polytope and under regularity assumptions, we prove complexity bounds which depend
on the combinatorial properties of . These bounds yield new
estimates on the complexity of solving -dim systems where all polynomials
share the same Newton polytope (\emph{unmixed case}). For instance, we
generalize the bound on the maximal degree in a Gr\"obner
basis of a -dim. bilinear system with blocks of variables of sizes
to the multilinear case: . We also propose
a variant of Fr\"oberg's conjecture which allows us to estimate the complexity
of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan
(2014
Source Galerkin Calculations in Scalar Field Theory
In this paper, we extend previous work on scalar theory using the
Source Galerkin method. This approach is based on finding solutions to
the lattice functional equations for field theories in the presence of an
external source . Using polynomial expansions for the generating functional
, we calculate propagators and mass-gaps for a number of systems. These
calculations are straightforward to perform and are executed rapidly compared
to Monte Carlo. The bulk of the computation involves a single matrix inversion.
The use of polynomial expansions illustrates in a clear and simple way the
ideas of the Source Galerkin method. But at the same time, this choice has
serious limitations. Even after exploiting symmetries, the size of calculations
become prohibitive except for small systems. The calculations in this paper
were made on a workstation of modest power using a fourth order polynomial
expansion for lattices of size ,, in , , and . In
addition, we present an alternative to the Galerkin procedure that results in
sparse matrices to invert.Comment: 31 pages, latex, figures separat
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