The Pythagoras number and the u-invariant of Laurent series fields in several variables

JOURNAL OF ALGEBRA

卷:426

Article

Hu, Yong

关键词: Quadratic forms;    Pythagoras number;    u-Invariant;    Sums of squares;    Laurent series fields;   

DOI  :  10.1016/j.jalgebra.2014.11.026

来源: Elsevier

【 摘 要 】

We show that every sum of squares in the three-variable Laurent series field R((x, y, z)) is a sum of 4 squares, as was conjectured in a paper of Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that every sum of squares in a finite extension of R((x, y)) is a sum of 3 squares. It was already shown in Choi, Dai, Lam and Reznick's paper that every sum of squares in R((x, y)) itself is a sum of two squares. We give a generalization of this result where R is replaced by an arbitrary real field. Our methods yield similar results about the u-invariant of fields of the same type. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free