caa a22 4500 606199691 CHVBK 20210128100939.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s11045-013-0265-0 doi (NATIONALLICENCE)springer-10.1007/s11045-013-0265-0 Pommaret Jean-Francois CERMICS, Ecole des Ponts ParisTech, 6/8 Av. Blaise Pascal, 77455, Marne-la-Vallée Cedex 02, France aut Relative parametrization of linear multidimensional systems [Elektronische Daten] [Jean-Francois Pommaret] 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 $$0$$ 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 $$r$$ r -pure module can be embedded into a module of projective dimension $$r$$ r , that is a module admitting a projective resolution with exactly $$r$$ 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. Springer Science+Business Media New York, 2013 Unmixed polynomial ideal nationallicence Extension module nationallicence Torsion-free module nationallicence Pure differential module nationallicence Involution nationallicence Spencer operator nationallicence Inverse system nationallicence Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 405-437 0923-6082 26:2<405 2015 26 11045 https://doi.org/10.1007/s11045-013-0265-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s11045-013-0265-0 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Pommaret Jean-Francois CERMICS, Ecole des Ponts ParisTech, 6/8 Av. Blaise Pascal, 77455, Marne-la-Vallée Cedex 02, France aut NATIONALLICENCE 773 0- Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 405-437 0923-6082 26:2<405 2015 26 11045