Journal of Pure and Applied Algebra 193 (2004) 307 – 311
www.elsevier.com/locate/jpaa
An elementary proof that the length of
X
4
1
+ X
4
2
+ X
4
3
+ X
4
4
is 4
Digen Zhang
∗
Department of Mathematics, University of Regensburg, Regensburg 93040, Germany
Received 7 April 2003; received in revised form2 February 2004
Communicated by M.-F. Roy
Abstract
An elementary proof is given, to show that the quartic form X
4
1
+ X
4
2
+ X
4
3
+ X
4
4
cannot be
written as a sumof three squares of real quadratic forms.
c
2004 Elsevier B.V. All rights reserved.
MSC: 11E25
1. Introduction
Reznick [3] conjectured that in the polynomial ring A := R[X
1
;X
2
;:::;X
2
k
]in2
k
indeterminates, the length of the form
S
k
:= X
2
k
1
+ X
2
k
2
+ ···+ X
2
k
2
k
is 2
k
, i.e., the form S
k
=
2
k
i=1
X
2
k
i
cannot be written as a sumof fewer than 2
k
squares
of homogeneous polynomials over R. This conjecture is evidently true for k = 1. Yiu
[5] establish its validity for k =2:
Theorem 0 (cf. Yiu [5]). The quartic form X
4
1
+ X
4
2
+ X
4
3
+ X
4
4
cannot be written as
a sum of three squares of real quadratic forms.
The proof of the above theoremin [5] is a geometric argument. The purpose of this
note is to give an elementary proof.
∗
Tel.: +49-941-9432764; fax: +49-941-9432576.
E-mail address: digen.zhang@mathematik.uni-regensburg.de (D. Zhang).
0022-4049/$ - see front matter
c
2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2004.02.012