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Algorithms in SINGULAR: Parallelization, Syzygies, and Singularities

  • This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts. The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm. Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations. The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research. In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused. Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.

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Author:Andreas Steenpass
URN (permanent link):urn:nbn:de:hbz:386-kluedo-38405
Advisor:Wolfram Decker
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2014/07/24
Year of Publication:2014
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2014/07/24
Date of the Publication (Server):2014/07/30
Number of page:XVII, 123
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):13-XX COMMUTATIVE RINGS AND ALGEBRAS / 13Bxx Ring extensions and related topics / 13B22 Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
13-XX COMMUTATIVE RINGS AND ALGEBRAS / 13Dxx Homological methods (For noncommutative rings, see 16Exx; for general categories, see 18Gxx) / 13D02 Syzygies, resolutions, complexes
13-XX COMMUTATIVE RINGS AND ALGEBRAS / 13Fxx Arithmetic rings and other special rings / 13F20 Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
14-XX ALGEBRAIC GEOMETRY / 14Bxx Local theory / 14B05 Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Wxx Algorithms (For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx) / 68W10 Parallel algorithms
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 10.09.2012