First presenter Co-presenter(s)
Name :  Alban Quadrat * Name:   
E-mail: E-mail:  
Affiliation: Institut National de Recherche en Informatique et en Automatique (INRIA) Name:   
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Country: France Name:   
10-04  E-mail:    
Session: 10- Algebraic and Algorithmic Aspects of Differential and Integral Operators Schedule:
Thursday, 16:00
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Title of
A Normal Form for 2-dimensional Linear Functional Systems
Slides of this talk (PDF)

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.