no. 1
On p-adic zeros of systems of diagonal forms restricted by a congruence condition
Godhino, Hemar1 ; Rodrigues, Paulo H. A.2
1 Departamento de Matemática Universidade de Brasília 70.910-900, Brasília, DF, Brasil
2 Instituto de Matemática e Estatística Universidade Federal de Goiás 74.001-970, Goiânia, GO, Brasil
Journal de théorie des nombres de Bordeaux, Tome 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 xyz≢0(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 p∤k. 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 xyz≢0(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 p∤k. L’hypothèse ci-dessus pour l’existence de solutions non-triviales de l’équation est vérifiée si, par exemple, p>k 4 .

DOI : 10.5802/jtnb.582

Godhino, Hemar 1 ; Rodrigues, Paulo H. A. 2

1 Departamento de Matemática Universidade de Brasília 70.910-900, Brasília, DF, Brasil
2 Instituto de Matemática e Estatística Universidade Federal de Goiás 74.001-970, Goiânia, GO, Brasil 
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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, Tome 19 (2007) no. 1, pp. 205-219. doi: 10.5802/jtnb.582

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