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

Hemar Godhino; Paulo H. A. Rodrigues

doi:10.5802/jtnb.582

On

p

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

Hemar Godhino ¹ ; Paulo H. A. Rodrigues 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.

Résumé
Abstract

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, $p > k^{4}$ .

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 $p > k^{4}$ .

MR Zbl

DOI : 10.5802/jtnb.582

Affiliations des auteurs :

Hemar Godhino ¹ ; Paulo H. A. Rodrigues 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

@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},
     pages = {205--219},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.582},
     mrnumber = {2332062},
     zbl = {1131.11023},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.582/}
}

TY  - JOUR
AU  - Hemar Godhino
AU  - Paulo H. A. Rodrigues
TI  - On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2007
SP  - 205
EP  - 219
VL  - 19
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.582/
DO  - 10.5802/jtnb.582
LA  - en
ID  - JTNB_2007__19_1_205_0
ER  -

%0 Journal Article
%A Hemar Godhino
%A Paulo H. A. Rodrigues
%T On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition
%J Journal de théorie des nombres de Bordeaux
%D 2007
%P 205-219
%V 19
%N 1
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.582/
%R 10.5802/jtnb.582
%G en
%F JTNB_2007__19_1_205_0

Hemar Godhino; Paulo H. A. Rodrigues. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.582/

Bibliographie
Cité par

[1] O.D. Atkinson, J. Brüdern, R.J. Cook, Simultaneous additive congruences to a large prime modulus. Mathematika 39 (1) (1992), 1–9. | MR | Zbl

[2] H. Davenport, D.J. Lewis, Simultaneous equations of additive type. Philos. Trans. Roy. Soc. London, Ser. A 264 (1969), 557–595. | MR | Zbl

[3] H. Godinho, P. H. A. Rodrigues, Conditions for the solvability of systems of two and three additive forms over p-adic fields. Proc. of the London Math. Soc. 91 (2005), 545–572. | MR | Zbl

[4] D.J. Lewis, H. Montgomery, On zeros of p-adic forms. Michigan Math. Journal 30 (1983), 83–87. | MR | Zbl

[5] L. Low, J. Pitman, A. Wolff, Simultaneous Diagonal Congruences. J. Number Theory 29 (1988), 31–59. | MR | Zbl

[6] I. D. Meir, Pairs of Additive Congruences to a Large Prime Modulus. J. Number Theory 63 (1997), 132–142. | MR | Zbl

Cité par Sources :