[Aalto AGC seminar] Vanni Noferini | Rosenbrock's Theorem characterizes Prüfer domains | Mon 05.10 at 14.15

[Aalto Algebra, Geometry and Combinatorics Seminar]

We would like to invite you to the next talk of autumn in the Aalto AGC (Algebra, Geometry and Combinatorics) Seminar!

The talk is on
Mon 05.10. at 14:15-15:00
in
M3 (M234), Otakaari 1.

The speaker is
Vanni Nocerini (Aalto)
giving the talk
Rosenbrock's Theorem characterizes Prüfer domains

Abstract:
Under coprimality assumptions on certain submatrices, Rosenbrock's Theorem relates the invariant factors of a matrix P to those of its "transfer function" (an engineering term that denotes the Schur complement of a leading principal submatrix, assumed invertible) G. Motivated by applications in systems theory, Rosenbrock first proved this result in the 1960s for P \in \R[x]. In 1974, Coppel proved Rosenbrock's Theorem in much broader generality, assuming that the base ring is a PID.

I would like to discuss some recent results on this theorem:


  1.
Even Coppel's assumptions can be weakened further to show that Rosenbrock's Theorem holds over any elementary divisor domain (EDD). This is relevant for engineering applications over the ring of entire functions. Moreover, even when the assumptions of the theorem do not hold, it is possible to estimate the relation between the invariant factors of P and G.
  2.
Because the Smith and Smith-McMIllan forms are only guaranteed to exist over an EDD (or its field of fractions), the result in part (a) means that Rosenbrock's Theorem holds for every integral domain over which its original statement makes sense. However, it possible to reopen the question by translating the statement in the language of fractional ideals. One can then prove that this "ideal-theoretic Rosenbrock's Theorem" holds if and only if the underlying ring is a Prüfer domain. For more general rings, several interesting partial results can also be stated, some of which are still motivated by applications to engineering. Some connections with the Quillen-Suslin theorem also arise.



Part (a) is based on joint work with Dopico and Zaballa; part (b) is based on a single authored paper.


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