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

Godhino, Hemar, and 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 19.1 (2007): 205-219. <http://eudml.org/doc/249965>.

@article{Godhino2007,
abstract = {This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $ax^k+by^k+cz^k \equiv d \,(\mbox \{mod\}\,p)$ has a solution with $xyz \nequiv0 \,(\mbox \{mod\}\,p)$ 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 $p \nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p &gt; k^4$.},
affiliation = {Departamento de Matemática Universidade de Brasília 70.910-900, Brasília, DF, Brasil; Instituto de Matemática e Estatística Universidade Federal de Goiás 74.001-970, Goiânia, GO, Brasil},
author = {Godhino, Hemar, Rodrigues, Paulo H. A.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Diagonal forms; -adic zeros},
language = {eng},
number = {1},
pages = {205-219},
publisher = {Université Bordeaux 1},
title = {On $\{p\}$-adic zeros of systems of diagonal forms restricted by a congruence condition},
url = {http://eudml.org/doc/249965},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Godhino, Hemar
AU - Rodrigues, Paulo H. A.
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
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 205
EP - 219
AB - This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $ax^k+by^k+cz^k \equiv d \,(\mbox {mod}\,p)$ has a solution with $xyz \nequiv0 \,(\mbox {mod}\,p)$ 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 $p \nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p &gt; k^4$.
LA - eng
KW - Diagonal forms; -adic zeros
UR - http://eudml.org/doc/249965
ER -