Parametrising the solutions of the generalised partial realisation problem through an atlas of coordinates

Abstract

Given a finite nice sequence of matrices ℒ and a positive integer d, the set of generalised partial realisations of ℒ of order d is a stratified smooth manifold. An atlas of coordinate charts for each stratum is provided that locally parametrises it. In particular, the set of all generalised partial realisations of order d of any given nice sequence ℒ will be described.

Keywords:

• partial realisation
• brunovsky indices
• reduced form
• smooth manifold
• atlas of coordinates
• parametrisation

AMS Subject Classifications::

• 93B15
• 93B29
• 93C05

Acknowledgements

The authors are grateful to anonymous referees whose suggestions improved this article. I. Baragaña was partially supported by MICINN MTM2010-19356-C02-01, EJ GIC10/169-IT-361-10. F. Puerta was partially supported by MICINN MTM2011-23892 (J-01698). I. Zaballa was partially supported by MICINN MTM2010-19356-C02-01, EJ GIC10/169-IT-361-10 and UPV/EHU UFI11/52.