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Macaulay inverse systems revisited
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
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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