Bien qu’elles aient une infinité de solutions, on peut voir les équations de Pell-Fermat comme des ancêtres des équations de Thue. L’analogie se resserre lorsqu’on les étudie sur les anneaux de polynômes en caractéristique nulle. Nous poursuivons l’étude entreprise par D. Masser et U. Zannier dans ce cadre, en considérant le cas de discriminants admettant une racine double.
Pell equations over the ring of integers are the forerunners of Thue equations. In fact, they too often have only finitely many solutions, when set over polynomial rings in characteristic zero. How often this happens has been the theme of recent work of D. Masser and U. Zannier. We pursue this study by considering Pell equations with non square-free discriminants over such rings.
Mots-clés : affine singular curves, generalized jacobians, Manin-Mumford conjecture, polynomial Pell equations
@article{JTNB_2015__27_2_439_0,
author = {Bertrand, Daniel},
title = {Generalized jacobians and {Pellian} polynomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {439--461},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {27},
number = {2},
year = {2015},
doi = {10.5802/jtnb.909},
mrnumber = {3393162},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.909/}
}
TY - JOUR AU - Bertrand, Daniel TI - Generalized jacobians and Pellian polynomials JO - Journal de théorie des nombres de Bordeaux PY - 2015 SP - 439 EP - 461 VL - 27 IS - 2 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.909/ DO - 10.5802/jtnb.909 LA - en ID - JTNB_2015__27_2_439_0 ER -
%0 Journal Article %A Bertrand, Daniel %T Generalized jacobians and Pellian polynomials %J Journal de théorie des nombres de Bordeaux %D 2015 %P 439-461 %V 27 %N 2 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.909/ %R 10.5802/jtnb.909 %G en %F JTNB_2015__27_2_439_0
Bertrand, Daniel. Generalized jacobians and Pellian polynomials. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 439-461. doi : 10.5802/jtnb.909. http://www.numdam.org/articles/10.5802/jtnb.909/
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