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227
On Fast Multiplication of Polynomials Over Arbitrary Algebras
 Acta Informatica
, 1991
"... this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polyno ..."
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Cited by 151 (6 self)
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this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polynomials of degree ! n in
Diagonal Algebraic SpaceTime Block Codes
 IEEE Trans. Inform. Theory
"... We construct a new family of linear spacetime block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasistatic or fast fading channels. The proposed codes transmit at a normalized rate of 1 symbol/sec. ..."
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Cited by 86 (7 self)
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We construct a new family of linear spacetime block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasistatic or fast fading channels. The proposed codes transmit at a normalized rate of 1 symbol/sec. When the number of transmit antennae n =1, 2 or n is a multiple of 4 we spread a rotated version of the information symbol vector by the Hadamard transform and send it over n transmit antennae and n time periods; for other values of n, we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an nn spacetime code matrix. The codes maintain their rate, diversity and coding gains for all real and complex constellations carved from the complex integers ring Z[i], and they outperform the codes from orthogonal design when using complex constellations for n > 2. The maximum likelihood decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity. It is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity.
Invariant measures for higherrank hyperbolic abelian actions
, 2002
"... We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of R k, Z k and Z k +. We show that they are either Haar measures or that every element of the action has zero metric entropy. ..."
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Cited by 72 (26 self)
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We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of R k, Z k and Z k +. We show that they are either Haar measures or that every element of the action has zero metric entropy.
Implementing 2Descent for Jacobians of Hyperelliptic Curves
 Acta Arith
, 1999
"... . This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one w ..."
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Cited by 47 (16 self)
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. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the MordellWeil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...
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 42 (4 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.
A W Reid, The Bianchi groups are separable on geometrically finite subgroups
 Ann. of Math
, 2001
"... Let d be a square free positive integer and Od the ring of integers in Q ( √ −d). The main result of this paper is to show that the groups PSL(2, Od) are subgroup separable on geometrically finite subgroups. 1 ..."
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Cited by 33 (8 self)
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Let d be a square free positive integer and Od the ring of integers in Q ( √ −d). The main result of this paper is to show that the groups PSL(2, Od) are subgroup separable on geometrically finite subgroups. 1
Sharp estimates for the arithmetic Nullstellensatz
 Duke Math. J
"... We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which ..."
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Cited by 25 (2 self)
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We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine
Visible evidence in the Birch and SwinnertonDyer Conjecture for modular abelian varieties of analytic rank zero
, 2004
"... This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic ..."
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Cited by 23 (15 self)
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This paper provides evidence for the Birch and SwinnertonDyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of �(Af). We find that there are at least 168 such Af for which the Birch and SwinnertonDyer conjecture implies that �(Af) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of �(Af) really divides # �(Af) by constructing nontrivial elements of �(Af) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of ShafarevichTate groups of elliptic curves.
An elementary problem equivalent to the Riemann hypothesis
 Amer. Math. Monthly
"... ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn), ..."
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Cited by 21 (2 self)
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ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),