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18
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
On NPPartitions over Posets with an Application to Reducing the Set of Solutions of NP Problems
 In Proceedings 25th Symposium on Mathematical Foundations of Computer Science
, 2000
"... . The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartiti ..."
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Cited by 12 (3 self)
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. The boolean hierarchy of kpartitions over NP for k 2 was introduced as a generalization of the wellknown boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NPpartitions by considering partition classes which are generated by nite labeled posets. Since we cannot prove it absolutely, we collect evidence for this extended boolean hierarchy to be strict. We give an exhaustive answer to the question of which relativizable inclusions between partition classes can occur depending on the relation between their dening posets. The study of the extended boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions of NP problems. For nite cardinality types, assuming the extended boolean hierarchy of kpartitions over NP is strict, we give a complete characterization when such solution reductions are possible. 1 Introduct...
Algorithms for Matrix Groups and the Tits Alternative
 Proc. 36th IEEE FOCS
, 1999
"... l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals ..."
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Cited by 11 (2 self)
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l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 00220000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences
Computing the maximal subgroups of a permutation group I
, 2001
"... We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach ..."
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Cited by 5 (2 self)
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We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach
Polynomialtime computation in matrix groups
, 1999
"... This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that e ..."
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Cited by 5 (3 self)
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This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that enable such efficiency. A
An Invitation to Computational Group Theory
 Groups' 93  Galway/St. Andrews, volume 212 of London Math. Soc. Lecture Note Ser
, 1995
"... Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (19671977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some ..."
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Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (19671977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases varied and improved, work horses of CGT systems: Sims' methods for handling big permutation groups [Sim70], the KnuthBendix method for attempting to construct a rewrite system from a presentation [KB70], variations of the ToddCoxeter method for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the pNilpotentQuotient method of I. D. Macdonald [Mac74] and the ReidemeisterSchreier method of G. Havas [Hav74] for subgroup presentations were published within a few years from that conference. However at least equally important for making group theorists aware of CGT were a number of applications of...
Representing quotients of permutation groups
 Quart. J. Math. (Oxford
, 1997
"... IN this note, we consider the following problem. Let G be a finite permutation group of degree d, and let Nbea normal subgroup of G. Under what circumstances does G/N have a faithful permutation representation of degree at most di ..."
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IN this note, we consider the following problem. Let G be a finite permutation group of degree d, and let Nbea normal subgroup of G. Under what circumstances does G/N have a faithful permutation representation of degree at most di
PolynomialTime Normalizers for Permutation Groups With Restricted Composition Factors
, 2002
"... For an integer constant d > 0, let d denote the class of finite groups all of whose nonabelian composition factors lie in S d ; in particular, d includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation fo ..."
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For an integer constant d > 0, let d denote the class of finite groups all of whose nonabelian composition factors lie in S d ; in particular, d includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, setstabilizers, group intersections, and centralizers have all been shown to be polynomialtime computable. The most notable gap in the theory has been the question of whether normalizers of subgroups can be found in polynomial time. We resolve this question in the affirmative. Among other new procedures, the algorithm requires instances of subspacestabilizers for certain linear representations and therefore some polynomialtime computation in matrix groups.