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.

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $a{x}^{k}+b{y}^{k}+c{z}^{k}\equiv d\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}p\right)$ has a solution with $xyz\not\equiv 0\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}p\right)$ 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 $p\nmid k$. 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 $a{x}^{k}+b{y}^{k}+c{z}^{k}\equiv d\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}p\right)$ ait une solution telle que $xyz\not\equiv 0\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}p\right)$, 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 $p\nmid k$. L’hypothèse ci-dessus pour l’existence de solutions non-triviales de l’équation est vérifiée si, par exemple, $p>{k}^{4}$.

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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