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156
Mahler's Measure and Special Values of Lfunctions
, 1998
"... this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula, ..."
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Cited by 70 (1 self)
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this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,
Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 49 (35 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multilevel implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.
Primes In Elliptic Divisibility Sequences
"... Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical vie ..."
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Cited by 41 (13 self)
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Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.
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 39 (3 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.
On groups generated by two positive multitwists: Teichmüller curves and Lehmer’s number
, 2004
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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 35 (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
A family of pseudoAnosov braids with small dilatation
, 2006
"... This paper describes a family of pseudoAnosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudoAnosov braid with 2g + 1 strands determines a hyperelliptic mapping class with the same dilatation on a genus–g surface. P ..."
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Cited by 34 (13 self)
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This paper describes a family of pseudoAnosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudoAnosov braid with 2g + 1 strands determines a hyperelliptic mapping class with the same dilatation on a genus–g surface. Penner showed that logarithms of least dilatations of pseudoAnosov maps on a genus–g surface grow asymptotically with the genus like 1/g, and gave explicit examples of mapping classes with dilatations bounded above by log 11/g. Bauer later improved this bound to log 6/g. The braids in this paper give rise to mapping classes with dilatations bounded above by log(2 + √ 3)/g. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus–g surfaces.
Coxeter groups, Salem numbers and the Hilbert metric
, 2001
"... this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Le ..."
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Cited by 32 (6 self)
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this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Lehmer 1:1762808. Here Lehmer denotes Lehmer's number, a root of the polynomial 1 + x \Gamma x 3 \Gamma x 4 \Gamma x 5 \Gamma x 6 \Gamma x 7 + x 9 + x 10 (1.1) and the smallest known Salem number. Billiards. Recall that a Coxeter system (W; S) is a group W with a finite generating set S = fs 1 ; : : : ; s n g, subject only to the relations (s i s j ) m ij = 1, where m ii = 1 and m ij 2 for i 6= j. The permuted products s oe1 s oe2 \Delta \Delta \Delta s oen 2 W; oe 2 S n ; are the Coxeter elements of (W; S). We say w 2 W is essential if it is not conjugate into any subgroup W I ae W generated by a proper subset I ae S. The Coxeter group W acts naturally by reflections on V
A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions
 Duke Math. J
, 2003
"... Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K). ..."
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Cited by 25 (2 self)
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Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K).