68,377 research outputs found

    Macaulay inverse systems revisited

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    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.Comment: From a lecture at the International Conference : Application of Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI

    Differential equations associated to Families of Algebraic Cycles

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    We develop a theory of differential equations associated to families of algebraic cycles in higher Chow groups (i.e., motivic cohomology groups). This formalism is related to inhomogeneous Picard--Fuchs type differential equations. For families of K3 surfaces the corresponding non-linear ODE turns out to be symilar to Chazy's equation.Comment: 8 pages. Final version. To be published in Annales de l'Institute Fourie
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