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ON ARTIN'S CONJECTURE, II: PAIRS OF ADDITIVE FORMS

Published online by Cambridge University Press:  29 April 2002

J. BRÜDERN  and
H. GODINHO
J. BRÜDERN
Affiliation:
Mathematisches Institut A, Universität Stuttgart, D-70511 Stuttgart, Germany. [email protected]
H. GODINHO
Affiliation:
Departamento de Matemática, Universidade de Brasilia, Brasilia, DF 70910, Brazil. [email protected]
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Abstract

It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.

Keywords

Artin's conjectureadditive forms$p$-adic fields
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 84 , Issue 3 , May 2002 , pp. 513 - 538
Copyright
2002 London Mathematical Society

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