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On Artin's Conjecture, II: Pairs of Additive Forms
Author(s) -
Brüdern J.,
Godinho H.
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013588
Subject(s) - mathematics , conjecture , combinatorics , constant (computer programming) , pure mathematics , discrete mathematics , programming language , computer science
It is shown that the system of two additive equations a 1 x 1 k + … + a s x s k = b 1 x 1 k + … + b s x s k = 0 where k ⩾ 2 and a j , b j are any given integers, has non‐trivial solutions in all p ‐adic fields provided only that s > 8 k 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 8 k 2 can be replaced by 2 k 2 . 2000 Mathematical Subject Classification : 11D72.