MACAULAY INVERSE SYSTEMS REVISITED

Jean-François Pommaret (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal theory of systems of partial differential equations (Spencer operator on sections, involution,...) and its algebraic counterpart now called "algebraic analysis" (commutative and homological algebra, differential modules,...). Many explicit examples are fully treated and hints are given towards the way to work out computer algebra packages.

Mots clés: partial differential equations; Macaulay inverse systems; algebraic analysis; commutative algebra; homological algebra; localization; duality; computer algebra; Groebner bases.; Groebner bases

Domaines: Equations aux dérivées partielles [math.AP]; Algèbre commutative [math.AC]; Géométrie différentielle [math.DG]; Algèbres d'opérateurs [math.OA]

Jean-François Pommaret. MACAULAY INVERSE SYSTEMS REVISITED. Journal of Symbolic Computation, 2011, 46 (9), pp.1049-1069. ⟨10.1016/j.jsc.2011.05.007⟩. ⟨hal-00361230v2⟩