2,621 research outputs found
Primary ideals and their differential equations
An ideal in a polynomial ring encodes a system of linear partial differential
equations with constant coefficients. Primary decomposition organizes the
solutions to the PDE. This paper develops a novel structure theory for primary
ideals in a polynomial ring. We characterize primary ideals in terms of PDE,
punctual Hilbert schemes, relative Weyl algebras, and the join construction.
Solving the PDE described by a primary ideal amounts to computing Noetherian
operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms
for this task, and we present efficient implementations.Comment: 32 pages. To appear in Foundations of Computational Mathematic
An algorithm for primary decomposition in polynomial rings over the integers
We present an algorithm to compute a primary decomposition of an ideal in a
polynomial ring over the integers. For this purpose we use algorithms for
primary decomposition in polynomial rings over the rationals resp. over finite
fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos
to extract primary ideals from pseudo-primary ideals. A parallelized version of
the algorithm is implemented in SINGULAR. Examples and timings are given at the
end of the article.Comment: 8 page
Parallelization of Modular Algorithms
In this paper we investigate the parallelization of two modular algorithms.
In fact, we consider the modular computation of Gr\"obner bases (resp. standard
bases) and the modular computation of the associated primes of a
zero-dimensional ideal and describe their parallel implementation in SINGULAR.
Our modular algorithms to solve problems over Q mainly consist of three parts,
solving the problem modulo p for several primes p, lifting the result to Q by
applying Chinese remainder resp. rational reconstruction, and a part of
verification. Arnold proved using the Hilbert function that the verification
part in the modular algorithm to compute Gr\"obner bases can be simplified for
homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be
adapted to the local case, i.e. for local orderings and not necessarily
homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In
this paper we prove the corresponding theorem for non-homogeneous ideals in
case of a global ordering.Comment: 16 page
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