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36
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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Cited by 40 (3 self)
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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.
Keyexchange in real quadratic congruence function fields
 Designs, Codes and Cryptography 7
, 1996
"... ..."
The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
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Cited by 17 (2 self)
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Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
Approximating Rings of Integers in Number Fields
, 1994
"... In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be wellsolved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large ..."
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Cited by 16 (0 self)
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In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be wellsolved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tame" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a ...
Higher cohomology for abelian groups of toral automorphisms
 Ergod. Th. Dyn. Sys
, 1995
"... Abstract. In this note we extend the results of [3] which deal with description of smooth untwisted cohomology for Z kactions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found at one of the steps in the proof for the hyperbolic case. ..."
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Cited by 16 (8 self)
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Abstract. In this note we extend the results of [3] which deal with description of smooth untwisted cohomology for Z kactions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found at one of the steps in the proof for the hyperbolic case.
Decidability of the isomorphism problem for stationary AFalgebras and the associated ordered simple dimension groups, Ergodic Theory Dynam
 Systems
"... Abstract. The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C ∗isomorphism induces an equivalence relation on these matri ..."
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Cited by 13 (3 self)
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Abstract. The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C ∗isomorphism induces an equivalence relation on these matrices, called C ∗equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C ∗equivalent or not.
Equivalences Between Elliptic Curves and Real Quadratic Congruence Function Fields
 In preparation
"... In 1994, the wellknown DiffieHellman key exchange protocol was for the first time implemented in a nongroup based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a s ..."
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Cited by 13 (4 self)
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In 1994, the wellknown DiffieHellman key exchange protocol was for the first time implemented in a nongroup based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a socalled infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security of the protocol depends on a certain discrete logarithm problem (DLP). In this paper, we show that for real quadratic congruence function fields of genus one, i.e. elliptic congruence function fields, this DLP is equivalent to the DLP for elliptic curves over finite fields. We present the explicit corresponce between the two DLPs and prove some properties which have no analogues for real quadratic number fields. Furthermore, we show that for elliptic congruence function fields, the set of redu...
On the density of primes in arithmetic progression having a prescribed primitive root
, 1999
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Nonstationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams, Ergodic Theory Dynam. Systems
"... Abstract. We first study situations where the stable AFalgebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More gen ..."
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Cited by 7 (3 self)
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Abstract. We first study situations where the stable AFalgebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More generally we consider neccessary and sufficient conditions for two such matrices to determine isomorphic dimension groups. We give several examples. This paper was motivated by attempts in [BJO98] to classify certain AF algebras defined by constant incidence matrices. The specific incidence matrices considered in [BJO98] are of the form (18) below, and we shall see there that the first problem referred to in the abstract is most interesting for those matrices. The second problem referred to in the abstract is significant not only for AF algebras but also for e.g.classification of substitution minimal systems up to strong orbit equivalence, [GPS95], [For97], [DHS].homeomorphism classification of domains of certain inverse limit hyperbolic systems, [BD95], [SV98]. The latter paper, which was written independently of this paper, and which was pointed out to us by the referee, makes contributions