A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to modulo . We prove a similar result for polynomials that are divisible in by a polynomial of the form for some . We also formulate and prove an analogous statement for elliptic curves.
De nombreux auteurs ont prouvé des versions explicites de la conjecture de Lehmer dans le cas particulier de polynômes dont les coefficients sont tous congrus à modulo un entier . Nous prouvons ici un résultat similaire pour les polynômes qui sont divisibles dans l’anneau par un polynôme de la forme pour un certain . Nous prouvons également un énoncé analogue pour les courbes elliptiques.
Keywords: Lehmer conjecture, elliptic curve, canonical height
@article{JTNB_2012__24_3_751_0, author = {Silverman, Joseph H.}, title = {Lehmer{\textquoteright}s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {751--772}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {24}, number = {3}, year = {2012}, doi = {10.5802/jtnb.820}, zbl = {1264.11049}, mrnumber = {3010638}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.820/} }
TY - JOUR AU - Silverman, Joseph H. TI - Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 751 EP - 772 VL - 24 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.820/ DO - 10.5802/jtnb.820 LA - en ID - JTNB_2012__24_3_751_0 ER -
%0 Journal Article %A Silverman, Joseph H. %T Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves %J Journal de théorie des nombres de Bordeaux %D 2012 %P 751-772 %V 24 %N 3 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.820/ %R 10.5802/jtnb.820 %G en %F JTNB_2012__24_3_751_0
Silverman, Joseph H. Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 751-772. doi : 10.5802/jtnb.820. http://archive.numdam.org/articles/10.5802/jtnb.820/
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