Dissertation Non-Commutative Gr¨obnerBases and Applications Xingqiang Xiu Eingereicht an der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau als Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften Submitted to the Department of Informatics and Mathematics of the Universit¨atPassau in Partial Fulfilment of the Requirements for the Degree of a Doctor in the Domain of Science Betreuer / Advisor: Prof. Dr. Martin Kreuzer Universit¨atPassau May 2012 Non-Commutative Gr¨obnerBases and Applications Xingqiang Xiu Erstgutachter: Prof. Dr. Martin Kreuzer Zweitgutachter: Prof. Dr. Gerhard Rosenberger M¨undliche Pr¨ufer: Prof. Dr. Franz Brandenburg Prof. Dr. Tobias Kaiser Der Fakult¨atf¨urInformatik und Mathematik der Universit¨atPassau vorgelegt im Mai 2012 Dedicated to my parents and grandma ii Contents 1 Introduction 1 2 Preliminaries 7 2.1 Monoids and Groups . 7 2.2 Rings . 15 2.3 Modules . 21 3 Gr¨obnerBases in KhXi 27 3.1 Admissible Orderings . 28 3.2 The Division Algorithm . 34 3.3 Gr¨obnerBases . 40 3.4 Syzygies . 46 3.5 Gr¨obnerBases of Right Ideals . 52 4 Gr¨obnerBasis Computations in KhXi 55 4.1 The Buchberger Procedure . 57 4.2 Improved Buchberger Procedures . 68 4.2.1 Interreduction on Obstructions . 69 4.2.2 Improved Buchberger Procedures . 77 4.3 Homogenization and Dehomogenization . 96 4.4 Gr¨obnerBasis Computations for Right Ideals . 108 5 Gr¨obnerBasis Theory in (KhXi ⊗ KhXi)r 111 5.1 Module Term Orderings and the Division Algorithm . 112 5.2 Gr¨obnerBases and Gr¨obner Basis Computations . 118 iv CONTENTS 5.3 Improved Buchberger Procedures . 124 5.4 The F4 Procedure . 131 6 Applications 141 6.1 Gr¨obnerBases in KhXi=I and (KhXi=I ⊗ KhXi=I)r . 143 6.1.1 Gr¨obnerBases of Two-Sided Ideals in KhXi=I . 144 6.1.2 Gr¨obnerBases of Right Ideals in KhXi=I . 146 6.1.3 Gr¨obnerbases of Submodules in (KhXi=I ⊗ KhXi=I)r . 154 6.2 Elimination . 157 6.2.1 Elimination of Variables in KhXi . 157 6.2.2 Component Elimination in (KhXi ⊗ KhXi)r . 164 6.3 The K-Dimension of KhXi=I . 174 Bibliography 189 Chapter 1 Introduction In this thesis we devote ourselves to the study of non-commutative Gr¨obnerbases in free monoid rings over fields and in free bimodules over free monoid rings, and de- velop applications based on Gr¨obnerbases in these settings. In the past few decades, Gr¨obnerbases have had great success in computational commutative algebra and its applications. Moreover, Gr¨obnerbases and the Buchberger procedure for Gr¨obner basis computations have been extended successfully to various non-commutative algebras, and then found their ways into applications in those non-commutative settings. The computation of Gr¨obnerbases is a crucial point, both in theory and in practice. Con- sequently, one of the essential aims of this thesis is to develop efficient (enumerating) procedures for Gr¨obnerbasis computations. Motivation In 1965, B. Buchberger introduced Gr¨obnerbasis theory for ideals in commutative polynomial rings over fields (see [11]). He constructed special bases, named Gr¨obner bases, of ideals. A Gr¨obnerbasis G of an ideal is a set of polynomials such that every polynomial in the polynomial ring has a unique remainder when it is divided by the polynomials in G. In particular, the remainder of each polynomial in the ideal generated by G is zero. Buchberger developed a terminating procedure, called Buchberger's Algorithm, to transform a finite generating set of an ideal into a finite Gr¨obnerbasis of the same ideal. In a natural way, Gr¨obnerbases enable us to solve the membership problem for ideals, that is, to decide whether a given polynomial lies in a given ideal. Gr¨obnerbases also solve many other algebraic problems related to ideals in a computational fashion (see [14, 16, 18]). 2 1. Introduction Since it is outstandingly important for polynomial rings, Gr¨obner basis theory has been generalized to several algebraic structures. For instance, in their books [43, 44], M. Kreuzer and L. Robbiano describe a more general version of Gr¨obner basis theory for free modules over commutative polynomial rings, and provide numerous character- izations and applications of Gr¨obnerbases. A concept of Gr¨obnerbases for non-commutative polynomial rings (free monoid rings) over fields was first proposed by F. Mora [53], who formulated Buchberger's Algorithm to compute Gr¨obnerbases of ideals in non-commutative polynomial rings. The most important difference is that, non-commutative polynomial rings are no longer Noetherian if they are generated by more then one indeterminate. Hence the procedure for Gr¨obnerbasis computations may not terminate. Further, T. Mora [55] unified Gr¨obnerbasis theory for both commutative and non-commutative algebras. Originally, Gr¨obnerbasis theory was established by a rewriting approach, which uses polynomials as rewriting rules (see [11]). K. Madlener et al. adopted this method and defined the theory of the prefix Gr¨obnerbases in monoid and group rings (see [52, 57, 58, 63]). Aided by the development of computer algebra systems, Buchberger's Algorithm for computing Gr¨obner bases in commutative algebras has been improved and refined over several decades (see [4, 12, 13, 17, 28, 33, 34]). Today there is an implementation of Buchberger's Algorithm in virtually every computer algebra system, including CoCoA [20], GAP [31], Magma [51], SINGULAR [65], et cetera. However, there are only a few computer algebra systems providing a user with the possibility of performing computations in the non-commutative case. Besides ApCoCoA [2], we refer to [67], Section 5 for an exhaustive list of such systems. In this thesis we start by following the approach of [43] to characterize Gr¨obnerbases in free monoid rings over fields using the notions given by Mora [53, 55] (see Chapter 3). In full detail, we formulate an enumerating procedure, namely the Buchberger Procedure, for Gr¨obnerbasis computations and present several Improved Buchberger Procedures (see Chapter 4). Then, using the same approach, we investigate Gr¨obner basis theory in free bimodules over free monoid rings (see Chapter 5). Finally, in the last chapter (Chapter 6) we list a rich collection of useful applications of Gr¨obner bases. We want to mention that throughout the thesis we are adopting the notation and terminology of the books [43] and [44]. We have implemented all algorithms and procedures in this thesis in the package gbmr (the abbreviation for Gr¨obnerbases in monoid rings) of the computer algebra 3 system ApCoCoA. All examples provided in this thesis have been computed with this package gbmr. We refer to the ApCoCoA wiki page for more information on ApCoCoA and the package gbmr. Moreover, we refer to the Symbolic Data Project [68] for more examples of the applications of Gr¨obnerbases contributed by us. Outline This section presents an outline of the remainder of this thesis and our contributions to the topic at hand. Since every chapter starts with an explanation of its organization, we omit such descriptions here. Chapter 2 briefly introduces several basic algebraic categories. We need a number of definitions and notions from monoids and groups (see Section 2.1), rings (see Sec- tion 2.2) and modules (see Section 2.3). These are the basic algebraic objects in this thesis. Some important properties of these algebraic categories are reviewed. Moreover, the word problem, the membership problem and the conjugacy problem are defined. Chapter 3 introduces Gr¨obnerbases of ideals in free monoid rings and characterizes Gr¨obnerbases of ideals in detail. Gr¨obnerbases of ideals are defined with respect to a given admissible ordering σ as follows: given a two-sided ideal I, a subset G ⊆ I of non-zero polynomials is a σ-Gr¨obnerbasis of I if the leading term set LTσfGg = fLTσ(g) j g 2 Gg generates the leading term set LTσfIg = fLTσ(f) j f 2 Ig as a monoid ideal. Following the approach of M. Kreuzer and L. Robbiano in [43], we characterize Gr¨obnerbases via leading term sets and leading term ideals, Gr¨obner representations, and syzygy modules in great detail. In addition, Gr¨obnerbases of one-sided ideals are defined and characterized in the last section of Chapter 3. Chapter 4 focuses on techniques for Gr¨obnerbasis computations in free monoid rings. We check whether a set G of non-zero polynomials is a Gr¨obnerbasis via the set of obstructions of G: the set G is a Gr¨obner basis of the (two-sided) ideal it generates if the normal remainders with respect to G of all S-polynomials of obstructions are zero. Based on the idea of T. Mora [53, 55], we formulate a Buchberger Procedure to enumerate Gr¨obnerbases. However, in general there exist infinitely many obstructions for a given finite set of non-zero polynomials. This fact makes Buchberger's Procedure infeasible in practice. We get rid of a large number of trivial obstructions and for- mulate an improved version of the Buchberger Procedure which enumerates Gr¨obner bases for finitely generated ideals. Later, by investigating the set of non-trivial ob- structions carefully, we propose further improvements of the Buchberger Procedure by detecting unnecessary obstructions and by deleting redundant generators, respec- 4 1. Introduction tively. In order to detect as many unnecessary obstructions as possible, we present an Interreduction Algorithm on non-trivial obstructions and propose generalizations of the Gebauer-M¨uller Installation (see [33]) in free monoid rings. The effectiveness and efficiency of our improvements are shown in examples. Moreover, given a homogenous system of generators, we tune the Buchberger Procedure carefully and propose a ho- mogeneous version of the Buchberger Procedure to enumerate Gr¨obnerbases degree by degree.