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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
Displayed equations for Galois representations
The Galois representation associated to a p-divisible group over a complete
noetherian normal local ring with perfect residue field is described in terms
of its Dieudonn\'e display. As a corollary we deduce in arbitrary
characteristic Kisin's description of the Galois representation associated to a
commutative finite flat p-group scheme over a p-adic discrete valuation ring in
terms of its Breuil-Kisin module. This was obtained earlier by W. Kim by a
different method.Comment: 15 page
On a Ring of Formal Pseudo-differential Operators
We study the notion of non-commumative higher dimensional local fields. A
simplest example is the ring P of formal pseudo- differential operators. As an
application we extend the KP hierarchy to the space .Comment: LaTe
Macaulay inverse systems and Cartan-Kahler theorem
During the last months or so we had the opportunity to read two papers trying
to relate the study of Macaulay (1916) inverse systems with the so-called
Riquier (1910)-Janet (1920) initial conditions for the integration of linear
analytic systems of partial differential equations. One paper has been written
by F. Piras (1998) and the other by U. Oberst (2013), both papers being written
in a rather algebraic style though using quite different techniques. It is
however evident that the respective authors, though knowing the computational
works of C. done during the first half of the last century in a way not
intrinsic at all, are not familiar with the formal theory of systems of
ordinary or partial differential equations developped by D.C. Spencer
(1912-2001) and coworkers around 1965 in an intrinsic way, in particular with
its application to the study of differential modules in the framework of
algebraic analysis. As a byproduct, the first purpose of this paper is to
establish a close link between the work done by F. S. Macaulay (1862-1937) on
inverse systems in 1916 and the well-known Cartan-K{\"a}hler theorem (1934).
The second purpose is also to extend the work of Macaulay to the study of
arbitrary linear systems with variable coefficients. The reader will notice how
powerful and elegant is the use of the Spencer operator acting on sections in
this general framework. However, we point out the fact that the literature on
differential modules mostly only refers to a complex analytic structure on
manifolds while the Spencer sequences have been created in order to study any
kind of structure on manifolds defined by a Lie pseudogroup of transformations,
not just only complex analytic ones. Many tricky explicit examples illustrate
the paper, including the ones provided by the two authors quoted but in a quite
different framework
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