Diophantine inequalities with power sums
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, p. 547-560

The ring of power sums is formed by complex functions on of the form

α(n)=b1c1n+b2c2n+...+bhchn,

for some b i ¯ and c i . Let F(x,y) ¯[x,y] be absolutely irreducible, monic and of degree at least 2 in y. We consider Diophantine inequalities of the form

|F(α(n),y)|<|Fy(α(n),y)|·|α(n)|-ε

and show that all the solutions (n,y)× have y parametrized by some power sums in a finite set. As a consequence, we prove that the equation

F(α(n),y)=f(n),

with f[x] not constant, F monic in y and α not constant, has only finitely many solutions.

On appelle somme de puissances toute suite α: de nombres complexes de la forme

α(n)=b1c1n+b2c2n+...+bhchn,

où les b i ¯ et les c i sont fixés. Soit F(x,y) ¯[x,y] un polynôme unitaire, absolument irréductible, de degré au moins 2 en y. On démontre que les solutions (n,y)× de l’inégalité

|F(α(n),y)|<|Fy(α(n),y)|·|α(n)|-ε

sont paramétrées par un nombre fini de sommes de puissances. Par conséquent, on déduit la finitude des solutions de l’équation diophantienne

F(α(n),y)=f(n),

f[x] est un polynôme non constant et α 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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {2},
     year = {2007},
     pages = {547-560},
     doi = {10.5802/jtnb.601},
     mrnumber = {2394901},
     zbl = {1165.11036},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2007__19_2_547_0}
}
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://www.numdam.org/item/JTNB_2007__19_2_547_0/

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