Details:
Title  Macaulay inverse systems revisited  Author(s)  J.F. Pommaret  Type  Article in Journal  Abstract  Since its original publication in 1916 under the title The Algebraic Theory of Modular Systems, the book (Macaulay, 1916) by Macaulay has attracted a lot of scientists with a view towards pure mathematics (Eisenbud, 1996) or applications to control theory (Oberst, 1990) through the last chapter dealing with the socalled inverse system. The basic intuitive idea is the parallel existing between ideals in polynomial rings and systems of partial differential (PD) equations in one unknown with constant coefficients. A first purpose of this paper is thus to extend these results to arbitrary systems of PD equations by exhibiting a link with the formal theory of systems of PD equations (Pommaret, 1994; Seiler, 2009; Spencer, 1965) where concepts such as formal integrability and involution are superseding the Hbases of Macaulay. The second idea is to transfer the properties of ideals to their residue modules, in particular to extend to differential modules the unmixedness assumption of Macaulay. For this we use extensively the results of modern algebraic analysis (Bjork, 1993; Kashiwara, 1995; Palamodov, 1970; Pommaret, 2001, 2005), revisiting in particular the concept of purity by means of localization techniques. Accordingly, this paper can also be considered as a refinement and natural continuation of Pommaret (2007). Finally, following again Macaulay in the differential setting, the cornerstone and main novelty of the paper is to replace the socle of a module by the top of the corresponding dual system in order to be able to look for generators by using known arguments of algebraic geometry such as Nakayama’s lemma (Kunz, 1985). Many explicit examples are provided in order to illustrate the main constructive results that provide new hints for applying computer algebra to algebraic analysis (Quadrat, 2009). This paper is an extended version of a lecture given at the “Applications of Computer Algebra” meeting ACA 2008, held at RISCLinz, Austria, on July 27–30.  Keywords  Partial differential equations, Macaulay inverse system, Algebraic analysis, Commutative algebra, Homological algebra, Localization, Duality, Computer algebra, Gröbner bases  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717111000782 
Language  English  Journal  Journal of Symbolic Computation  Volume  46  Number  9  Pages  1049  1069  Year  2011  Edition  0  Translation 
No  Refereed 
No 
