A computational view on normal forms of matrices of Ore polynomials

Abstract

This thesis treats normal forms of matrices over rings of Ore polynomials. The whole thesis is divided in three parts: First, Ore polynomials are described and basic facts about them are recalled. This part also includes integro-differential operators as an extended example. Second, in the main part we present oneand two-sided normal forms of matrices. More precisely, we deal with the Popov normal form, Hermite normal form and the Jacobson normal form. In the last part, we explore an application of matrix normal forms to a problem in control theory. Below, we describe each of the parts in more detail. Ore polynomials, sometimes called skew polynomials, arise from the work of Oystein Ore in [Ore33]. They are a generalisation of the usual polynomials with almost all of their properties with the main exception being that the multiplication in not necessarily commutative: Neither need the coefficients commute with each other, nor does the indeterminate have to commute with them. Ore polynomials can be used to model differential or difference operators. For example, the famous Weyl algebra can be considered to be an Ore polynomial ring. As an example, we model integro-differential operators with polynomial coefficients using Ore polynomials. This part is based on our ISSAC 2009 paper [RRM09]. We arrive at a construction which is similar to the Weyl algebra in the purely differential case. In the main part, we consider normal forms of matrices. These make it possible to express systems of linear equations involving operators and to determine the properties of these systems such as, for example, solvability. We first consider normal forms with respect to row-operations. The coefficient domain here is a skew field. We treat row-reduction, the Hermite normal form, the Popov normal form and shifted Popov normal forms. We draw a connection between these normal forms to Grobner bases over modules. As an application of this connection, we present a modified FGLM algorithm for converting matrices from one normal form into another. Parts of this were presented at ACA 2010 and in [Mid10]. We also consider the Jacobson normal form which is a normal form with respect to simultaneous rowand column-operations. Here, we restrict ourselves to differential operators over a commutative coefficient domain. We present a modular algorithm for computing a Jacobson normal form which is based on cyclic vectors and which is guaranteed to succeed in characteristic zero, but under certain conditions also yields a result in positive characteristic. The last part deals with a topic from control theory. We examine linear time-varying differential systems with delays for differential flatness and π-flatness where we use an idea from [MCL10]. For this, we apply the one-sided normal forms from the main part instead of the originally proposed