The ring of power sums is formed by complex functions on of the form
for some and . Let be absolutely irreducible, monic and of degree at least in . We consider Diophantine inequalities of the form
and show that all the solutions have parametrized by some power sums in a finite set. As a consequence, we prove that the equation
with not constant, monic in and not constant, has only finitely many solutions.
On appelle somme de puissances toute suite de nombres complexes de la forme
où les et les sont fixés. Soit un polynôme unitaire, absolument irréductible, de degré au moins en . On démontre que les solutions de l’inégalité
sont paramétrées par un nombre fini de sommes de puissances. Par conséquent, on déduit la finitude des solutions de l’équation diophantienne
où est un polynôme non constant et est une somme de puissances non constante.
@article{JTNB_2007__19_2_547_0, author = {Scremin, Amedeo}, title = {Diophantine inequalities with power sums}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {547--560}, publisher = {Universit\'e Bordeaux 1}, volume = {19}, number = {2}, year = {2007}, doi = {10.5802/jtnb.601}, zbl = {1165.11036}, mrnumber = {2394901}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.601/} }
TY - JOUR AU - Scremin, Amedeo TI - Diophantine inequalities with power sums JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 547 EP - 560 VL - 19 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.601/ DO - 10.5802/jtnb.601 LA - en ID - JTNB_2007__19_2_547_0 ER -
Scremin, Amedeo. Diophantine inequalities with power sums. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 547-560. doi : 10.5802/jtnb.601. http://archive.numdam.org/articles/10.5802/jtnb.601/
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