Generalized jacobians and Pellian polynomials
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, p. 439-461

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.

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.

DOI : https://doi.org/10.5802/jtnb.909
Classification:  14H25,  11G30,  14D10
Keywords: 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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     pages = {439-461},
     doi = {10.5802/jtnb.909},
     mrnumber = {3393162},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2015__27_2_439_0}
}
Bertrand, Daniel. Generalized jacobians and Pellian polynomials. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 439-461. doi : 10.5802/jtnb.909. http://www.numdam.org/item/JTNB_2015__27_2_439_0/

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