On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition

Godhino, Hemar; Rodrigues, Paulo H. A.

doi:10.5802/jtnb.582

p

-adic zeros of systems of diagonal forms restricted by a congruence condition

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

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 205-219.

Abstract
Résumé

This paper is concerned with non-trivial solvability in $p$ -adic integers of systems of additive forms. Assuming that the congruence equation $a x^{k} + b y^{k} + c z^{k} \equiv d (m o d p)$ has a solution with $x y z ≢ 0 (m o d p)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2 \cdot 3^{R - 1} \cdot k + 1$ 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 $a x^{k} + b y^{k} + c z^{k} \equiv d (m o d p)$ ait une solution telle que $x y z ≢ 0 (m o d p)$ , nous montrons qu’un système quelconque de formes additives de degré $k$ et d’au moins $2 \cdot 3^{R - 1} \cdot k + 1$ 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, $p > k^{4}$ .

MR Zbl

DOI: 10.5802/jtnb.582

Author's affiliations:

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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     author = {Godhino, Hemar and Rodrigues, Paulo H. A.},
     title = {On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {205--219},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.582},
     zbl = {1131.11023},
     mrnumber = {2332062},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.582/}
}

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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, Volume 19 (2007) no. 1, pp. 205-219. doi : 10.5802/jtnb.582. http://archive.numdam.org/articles/10.5802/jtnb.582/

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