TY - JOUR
T1 - Quintic forms over p-adic fields
AU - Leep, David B.
AU - Yeomans, Charles C.
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 1996/4
Y1 - 1996/4
N2 - We prove that a quintic form in 26 variables defined over a p-adic field K always has a non trivial zero over K if the residue class field of K has at least 47 elements. This is in agreement with the theorem of Ax-Kochen which states that a homogeneous form of degree d in d2 + 1 variables defined over Qp has a nontrivial Qp-rational zero if p is sufficiently large. The Ax-Kochen theorem gives no results on the bound for p. For d= 1, 2, 3 it has been known for a long time that there is a nontrivial Qp-rational zero for all values of p. For d=4, Terjanian gave an example of a form in 18 variables over Q2 having no nontrivial Q2-rational zero. This is the first result which gives an effective bound for the case d= 5.
AB - We prove that a quintic form in 26 variables defined over a p-adic field K always has a non trivial zero over K if the residue class field of K has at least 47 elements. This is in agreement with the theorem of Ax-Kochen which states that a homogeneous form of degree d in d2 + 1 variables defined over Qp has a nontrivial Qp-rational zero if p is sufficiently large. The Ax-Kochen theorem gives no results on the bound for p. For d= 1, 2, 3 it has been known for a long time that there is a nontrivial Qp-rational zero for all values of p. For d=4, Terjanian gave an example of a form in 18 variables over Q2 having no nontrivial Q2-rational zero. This is the first result which gives an effective bound for the case d= 5.
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U2 - 10.1006/jnth.1996.0046
DO - 10.1006/jnth.1996.0046
M3 - Article
AN - SCOPUS:0030120363
SN - 0022-314X
VL - 57
SP - 231
EP - 241
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 2
ER -