Slides of this talk (PDF)
Jacobson normal forms of matrices with entries in polynomial rings of
ordinary differential or difference operators with coefficients in a
skew field play an important role in the study of linear systems of
ordinary differential or recurrence equations. Unfortunately, they
generally do not exist for matrices with entries in noncommutative
polynomial rings in more than one variable, i.e., they cannot be used to
study linear systems of partial differential equations, differential
time-delay systems or multi-indexed recurrence equations.
The purpose of this talk is to show that every matrix over a
noncommutative polynomial ring in two independent variables admits an
upper triangular reduction formed by three diagonal blocks: the first
diagonal block defines the torsion-free part of the linear system, the
second one defines the 1-dimensional part and the third one defines the
0-dimensional part of the system. Hence, the corresponding linear system
can be integrated in cascade by first solving the 0-dimensional system,
then the 1-dimensional one and finally the parametrizable one. This
form for 2-dimensional linear systems generalizes the Jacobson normal
form for 1-dimensional linear systems. This normal form, based on
difficult results of module theory (e.g., pure submodules, purity
filtration) and homological algebra (e.g., extension functor,
ext_D^i(ext_D^i(M, D), D), Baer's extensions), can be computed by means
of Groebner or Janet basis techniques. |