Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 751-772.

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 1 modulo un entier m>1. Nous prouvons ici un résultat similaire pour les polynômes qui sont divisibles dans l’anneau (/m)[X] par un polynôme de la forme 1+X++X n pour un certain nϵdeg(f). Nous prouvons également un énoncé analogue pour les courbes elliptiques.

A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (/m)[X] by a polynomial of the form 1+X++X n for some nϵdeg(f). We also formulate and prove an analogous statement for elliptic curves.

DOI : 10.5802/jtnb.820
Classification : 11G05, 11G50, 11J97, 14H52
Mots clés : Lehmer conjecture, elliptic curve, canonical height
Joseph H. Silverman 1

1 Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA
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Joseph H. Silverman. Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 751-772. doi : 10.5802/jtnb.820. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.820/

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