Macaulay inverse systems revisited
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 so-called 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 H-bases 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 RISC-Linz, Austria, on July 27-30. (C) 2011 Elsevier Ltd. All rights reserved.
macaulay inverse system algebraic analysis localization computer algebra partial differential equations commutative algebra homological algebra duality grobner bases