Algorithms for noncommutative differential operators
Mark W. Giesbrecht, G. D. F. Reid, Yang Zhang · 2004
The aim of this work is to study some noncommutative differential operators. We take an algorithmic approach as well as further developing the mathematics, and design and analyse algorithms to solve fundamental problems. We also give applications to differential equations. First we consider how to factor skew polynomials. These are polynomials in a differential or difference operator. Using the eigenring method, we present algorithms for computing factorizations and least common left multiple decompositions of skew polynomials over F q(t), for a prime power q = pμ (here F q is the finite field with q elements). Our algorithms are effective in the skew polynomial ring F q (t)[ D ; σ, δ] (where D t = σ(t) D + δ(t)), for any automorphism σ and any σ-derivation δ of F q(t). Most importantly, these algorithms are the first to run in time polynomial in the degree of the input. In the second part of this thesis, we presents theory and algorithms for noncommutative Grobner bases in Poincare-Birkhoff-Witt extensions. These extension rings generalize the previous domains over which non-commutative Grobner bases have been applied. Our approach to noncommutative Grobner bases differs from previous work, which assumes that the coefficients are from a field or commutative ring. This is relevant for computations involving Cartan's theory of moving frames, and we explore these applications. In the third part of this thesis, we further our study of computations with moving frames, and extend the Rust-Riquier existence and uniqueness theory to analytic PDEs written in terms of moving frames of non-commuting partial differential operators. The main idea for the theoretical development is to use the commutation relations between the partial differential operators to place them in a standard order. This normalization is exploited to generalize the corresponding steps of the commuting Rust-Riquier Theory to the noncommutative case.