# Noncommutative computer algebra with applications in algebraic analysis

Andres, Wolf Daniel (Author); Zerz, Eva (Thesis advisor)

*Aachen / Publikationsserver der RWTH Aachen University (2013, 2014)* [Dissertation / PhD Thesis]

*Page(s): X, 190 S.*

Abstract

After giving a careful introduction to (noncommutative) computer algebra, we present an algorithm for the elimination of variables from arbitrary G-algebras via Gröbner bases if the Elimination Lemma is applicable. With this algorithm, we automate the process of checking for a certain necessary property of a subalgebra, finding an elimination ordering, performing the Gröbner basis computation and intersecting the result with the subalgebra. Then we apply this algorithm to the computation of preimages of (left) ideals under homomorphisms of G-algebras, resulting in an enhancement of Levandovskyy's approach. Moreover, we generalize an idea by Noro to solve the problem of intersecting an ideal in an Ore localization of a G-algebra with a principal subalgebra, provided that the intersection is a priori known to be nontrivial. Further, we study the zeroth graded component of Kashiwara's and Malgrange's V-filtration on the n-th Weyl algebra. Under the condition, that the considered weight vector has identical entries, we show, that this component viewed as an algebra is isomorphic to a quotient of the universal enveloping algebra of the general linear Lie algebra and we determine explicitly the set of generators of the respective ideal. We also establish the Gel'fand-Kirillov dimension of the zeroth graded component by two distinct methods, a computer algebraic one and a ring theoretic one. Then we discuss operators satisfying Bernstein's functional equation and derive a necessary, purely commutative criterion for the existence of a Bernstein operator of a given order. We present and enhance a new method to compute Bernstein data, which, in contrast to the existing approaches, relies only on standard basis computations in commutative rings. Moreover, we require only about half of the number of variables compared to the known algorithms and thus, we seriously reduce the theoretic complexity of the computation of the Bernstein-Sato polynomial. In addition, we are able to treat the global and local situations almost identically. We convert our approach to compute Bernstein data to obtain a method to compute s-parametric annihilators up to a specified order. Another modification allows to compute the whole tower of annihilators up to a given order. These two algorithms also work over a commutative polynomial ring. Further, we give a general idea for a proof to a conjecture by Ucha-Enriquez and point out the missing link for a special case by elementary means. We also take a step towards a general solution by formulating (and proving) a general formula for the action of a linear differential operator with polynomial coefficients on products of symbolic powers of polynomials. By applying our results concerning the zeroth graded component of the V-filtration, we obtain an enhanced model for the D-module setting for affine algebraic varieties, which again reduces the theoretical complexity of the known approaches by a huge amount.Finally, we utilize Tsai's algorithm for the computation of the Weyl closure to construct a new algorithm to compute full s-parametric annihilators as well as one to compute the annihilator of the exponential functions of the inverse of a polynomial, which up to now seems to be the only known general method to solve this task.

### Identifier

- URN: urn:nbn:de:hbz:82-opus-49285
- REPORT NUMBER: RWTH-CONV-143884