NDLI: Relative parametrization of linear multidimensional systems

Content Provider	Springer Nature Link
Author	Pommaret, Jean Francois
Copyright Year	2013
Abstract	In the last chapter of his book “The Algebraic Theory of Modular Systems” published in 1916, F. S. Macaulay developped specific techniques for dealing with “unmixed polynomial ideals” by introducing what he called “inverse systems”. The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential or partial differential equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients. The first and main idea is to replace unmixed polynomial ideals by pure differential modules. The second idea is to notice that a module is $$0$$ -pure if and only if it is torsion-free and thus if and only if it admits an “absolute parametrization” by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an “absolute localization”. The third idea is to refer to a difficult theorem of algebraic analysis saying that an $$r$$ -pure module can be embedded into a module of projective dimension $$r$$ , that is a module admitting a projective resolution with exactly $$r$$ operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of “involution” and a ‘relative localization” leading to a “relative parametrization”. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.
Starting Page	405
Ending Page	437
Page Count	33
File Format	PDF
ISSN	09236082
Journal	Multidimensional Systems and Signal Processing
Volume Number	26
Issue Number	2
e-ISSN	15730824
Language	English
Publisher	Springer US
Publisher Date	2013-12-05
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Unmixed polynomial ideal Extension module Torsion-free module Pure differential module Involution Spencer operator Inverse system Circuits and Systems Electrical Engineering Signal, Image and Speech Processing Artificial Intelligence (incl. Robotics)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Artificial Intelligence Signal Processing Information Systems Computer Science Applications Software Hardware and Architecture

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