Displaying similar documents to “The integer transfinite diameter of intervals and totally real algebraic integers”

Zhang-Zagier heights of perturbed polynomials

Christophe Doche (2001)

Journal de théorie des nombres de Bordeaux

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In a previous article we studied the spectrum of the Zhang-Zagier height [2]. The progress we made stood on an algorithm that produced polynomials with a small height. In this paper we describe a new algorithm that provides even smaller heights. It allows us to find a limit point less than 1 . 289735 i.e. better than the previous one, namely 1 . 2916674 . After some definitions we detail the principle of the algorithm, the results it gives and the construction that leads to this new limit point. ...

On some subgroups of the multiplicative group of finite rings

José Felipe Voloch (2004)

Journal de Théorie des Nombres de Bordeaux

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Let S be a subset of F q , the field of q elements and h ∈ F q [ x ] a polynomial of degree d > 1 with no roots in S . Consider the group generated by the image of { x - s ∣ s ∈ S } in the group of units of the ring F q [ x ] / ( h ) . In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of...

On the binary expansions of algebraic numbers

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, Carl Pomerance (2004)

Journal de Théorie des Nombres de Bordeaux

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Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1 , then the number # ( | y | , N ) of 1-bits in the expansion of | y | through bit position N satisfies # ( | y | , N ) > C N 1 / D for a positive number C (depending on y ) and sufficiently large N . This in itself establishes the transcendency...

A new exceptional polynomial for the integer transfinite diameter of [ 0 , 1 ]

Qiang Wu (2003)

Journal de théorie des nombres de Bordeaux

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Using refinement of an algorithm given by Habsieger and Salvy to find integer polynomials with smallest sup norm on [0, 1] we extend their table of polynomials up to degree 100. For the degree 95 we find a new exceptionnal polynomial which has complex roots. Our method uses generalized Müntz-Legendre polynomials. We improve slightly the upper bound for the integer transfinite diameter of [0, 1] and give elementary proofs of lower bounds for the exponents of some critical polynomials. ...

New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

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Every real number x , 0 < x ≤ 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - ⋯ where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + ⋯ where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

On an approximation property of Pisot numbers II

Toufik Zaïmi (2004)

Journal de Théorie des Nombres de Bordeaux

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Let q be a complex number, m be a positive rational integer and l m ( q ) = inf { P ( q ) , P ∈ ℤ m [ X ] , P ( q ) ≠ 0 } , where ℤ m [ X ] denotes the set of polynomials with rational integer coefficients of absolute value ≤ m . We determine in this note the maximum of the quantities l m ( q ) when q runs through the interval ] m , m + 1 [ . We also show that if q is a non-real number of modulus > 1 , then q is a complex Pisot number if and only if l m ( q ) > 0 for all m .