95 research outputs found
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
Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules
Our recent arXiv preprints and published papers on the solution of the
Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the
importance of revisiting the algebraic structure of the Bianchi identities in
Riemannian geometry. We also discovered in the meantime that, in our first book
of 1978, we had already used a new way for studying the various compatibility
conditions (CC) of an operator that may not be necessarily formally integrable
(FI) in order to construct canonical formally exact differential sequences on
the jet level. The purpose of this paper is to prove that the combination of
these two facts clearly shows the specific importance of the Spencer operator
and the Spencer -cohomology, totally absent from mathematical physics
today. The results obtained are unavoidable because they only depend on
elementary combinatorics and diagram chasing. They also provide for the first
time the purely intrinsic interpretation of the respective numbers of
successive first, second, third and higher order generating CC. However, if
they of course agree with the linearized Killing operator over the Minkowski
metric, they largely disagree with recent publications on the respective
numbers of generating CC for the linearized Killing operator over the
Schwarzschild and Kerr metrics. Many similar examples are illustrating these
new techniques, providing in particular the only symbol existing in the
literature which is 2-ayclic witout being of finite type, contrary to the
conformal situation.Comment: This paper is achieving the work we started in arXiv:1805.11958 and
arXiv:2010.07001 (now published) on the search of the generating
compatibility conditions of the Killing operator for the Schwarzschild and
Kerr metrics by means of new intrinsic homological methods based on a
systematic use of the Spencer operato
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