ISBN 978-3-8439-3401-5

978-3-8439-3401-5, Reihe Mathematik

Sharwan Kumar Tiwari
Algorithms in Noncommutative Algebras: Gröbner Bases and Hilbert Series

107 Seiten, Dissertation Technische Universität Kaiserslautern (2017), Softcover, A5

Zusammenfassung / Abstract

This thesis deals with the efficient computation of Groebner bases over G-algebras defined over the rationals and of Hilbert series of finitely generated right modules over the free associative algebra. In particular, the thesis consists of two parts.

In the first part, we extend modular techniques for computing Groebner bases from the commutative setting to the vast class of noncommutative G-algebras. Similar to the commutative case, we prove an effective verification test for graded ideals. We have implemented our modular algorithm in the open source computer algebra system Singular and give timings to compare its performance with that of non-modular versions of Buchberger's algorithm in Singular. We consider test examples from D-module theory in the context of computing Bernstein-Sato polynomials together with some other classical benchmark examples.

In the second part of the thesis, we develop a well-tested implementation in the kernel of Singular to compute the univariate Hilbert series of noncommutative algebras based on the algorithm given by La Scala. Then we propose methods for computing multivariate Hilbert series of multigraded right modules over the free associative algebra. Using results from the theory of regular languages, we provide conditions under which the proposed methods are effective and hence the multivariate Hilbert series are rational functions with integer coefficients. In particular, we provide an implementation in the kernel of Singular for the computation of multivariate Hilbert series of noncommutative multigraded algebras. Furthermore, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, we also propose and implement efficient variants of the above-mentioned methods for computing (univariate and multivariate) Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We demonstrate the performance of the algorithms and their implementations for the computation of graded and multigraded Hilbert series, both in the complete and in the truncated case, on a number of research relevant examples. Moreover, we also consider some applications of the computation of multigraded Hilbert series to algebras which are invariant under the action of the general linear group.