Displaying similar documents to “On some subgroups of the multiplicative group of finite rings”

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

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Let M ( α ) be the Mahler measure of an algebraic number α , and V be an open subset of . Then its L ( V ) is inf M ( α ) 1 / deg ( α ) , the infimum being over all non-zero non-cyclotomic α lying with its conjugates outside V . We evaluate L ( V ) when V is any annulus centered at 0 . We do the same for a variant of L ( V ) , which we call the transfinite Lehmer constant L ( V ) .Also, we prove the converse to Langevin’s Theorem, which states that L ( V ) > 1 if V contains a point of modulus 1 . We prove the corresponding result for...

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 the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let K = 𝔽 q ( T ) be the rational function field over a finite field of q elements. For any polynomial f ( T ) 𝔽 q [ T ] with positive degree, denote by Λ f the torsion points of the Carlitz module for the polynomial ring 𝔽 q [ T ] . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K ( Λ P ) of degree k over 𝔽 q ( T ) , where P 𝔽 q [ T ] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q - 1 . A formula for the analytic class...

Linear independence of linear forms in polylogarithms

Raffaele Marcovecchio (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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For x , | x | < 1 , s , let Li s ( x ) be the s -th polylogarithm of x . We prove that for any non-zero algebraic number α such that | α | < 1 , the ( α ) -vector space spanned by 1 , Li 1 ( α ) , Li 2 ( α ) , has infinite dimension. This result extends a previous one by Rivoal for rational α . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

On coefficient valuations of Eisenstein polynomials

Matthias Künzer, Eduard Wirsing (2005)

Journal de Théorie des Nombres de Bordeaux

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Let p 3 be a prime, let n > m 1 . Let π n be the norm of ζ p n - 1 under C p - 1 , so that ( p ) [ π n ] | ( p ) is a purely ramified extension of discrete valuation rings of degree p n - 1 . The minimal polynomial of π n over ( π m ) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at π m . The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

Distribution of nodes on algebraic curves in N

Thomas Bloom, Norman Levenberg (2003)

Annales de l’institut Fourier

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Given an irreducible algebraic curves A in N , let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A . Let K be a nonpolar compact subset of A , and for each d = 1 , 2 , . . . , choose m d points { A d j } j = 1 , . . . , m d in K . Finally, let Λ d be the d -th Lebesgue constant of the array { A d j } ; i.e., Λ d is the operator norm of the Lagrange interpolation operator L d acting on C ( K ) , where L d ( f ) is the Lagrange interpolating polynomial for f of degree d at the points { A d j } j = 1 , . . . , m d . Using techniques...