On p-adic zeros of systems of diagonal forms restricted by a congruence condition
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Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 205-219.

This paper is concerned with non-trivial solvability in p-adic integers of systems of additive forms. Assuming that the congruence equation ax k +by k +cz k d(modp) has a solution with xyz0(modp) we have proved that any system of R additive forms of degree k with at least 2·3 R-1 ·k+1 variables, has always non-trivial p-adic solutions, provided pk. The assumption of the solubility of the above congruence equation is guaranteed, for example, if p>k 4 .

Cet article étudie l’existence de solutions non triviales en entiers p-adiques de systèmes d’équations pour des formes additives. En supposant que l’équation ax k +by k +cz k d(modp) ait une solution telle que xyz0(modp), nous montrons qu’un système quelconque de formes additives de degré k et d’au moins 2·3 R-1 ·k+1 variables possède toujours des solutions p-adiques non-triviales, si pk. L’hypothèse ci-dessus pour l’existence de solutions non-triviales de l’équation est vérifiée si, par exemple, p>k 4 .

Received: 2005-10-21
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.582
@article{JTNB_2007__19_1_205_0,
     author = {Hemar Godhino and Paulo H. A. Rodrigues},
     title = {On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {205-219},
     doi = {10.5802/jtnb.582},
     zbl = {1131.11023},
     mrnumber = {2332062},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_205_0/}
}
Godhino, Hemar; Rodrigues, Paulo H. A. On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 205-219. doi : 10.5802/jtnb.582. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_205_0/

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