En généralisant un résultat de Pourchet, nous démontrons que si sont deux sommes de puissances définies sur , satisfaisant certaines conditions nécessaires, la longueur de la fraction continue pour tend vers l’infini pour . On déduira ce résultat d’une inégalité de type Thue uniforme pour les approximations rationnelles des nombres de la forme .
Generalizing a result of Pourchet, we show that, if are power sums over satisfying suitable necessary assumptions, the length of the continued fraction for tends to infinity as . This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers , .
@article{JTNB_2005__17_3_737_0, author = {Corvaja, Pietro and Zannier, Umberto}, title = {On the length of the continued fraction for values of quotients of power sums}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {737--748}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {3}, year = {2005}, doi = {10.5802/jtnb.517}, mrnumber = {2212122}, zbl = {05016584}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.517/} }
TY - JOUR AU - Corvaja, Pietro AU - Zannier, Umberto TI - On the length of the continued fraction for values of quotients of power sums JO - Journal de Théorie des Nombres de Bordeaux PY - 2005 DA - 2005/// SP - 737 EP - 748 VL - 17 IS - 3 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.517/ UR - https://www.ams.org/mathscinet-getitem?mr=2212122 UR - https://zbmath.org/?q=an%3A05016584 UR - https://doi.org/10.5802/jtnb.517 DO - 10.5802/jtnb.517 LA - en ID - JTNB_2005__17_3_737_0 ER -
Corvaja, Pietro; Zannier, Umberto. On the length of the continued fraction for values of quotients of power sums. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 737-748. doi : 10.5802/jtnb.517. http://www.numdam.org/articles/10.5802/jtnb.517/
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