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 -