2013/03/31 by Yong Hu · 13 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Combinatorics #Discrete mathematics #Explained sum of squares #Extension (predicate logic) #Finite field #Generalization #Invariant (physics) #Lack-of-fit sum of squares #Laurent series #Least-squares function approximation #Mathematical analysis #Mathematical physics #Mathematics #Non-linear least squares #Polynomial and algebraic computation #Pure mathematics #Series (stratigraphy) #Statistics #math.NT #msc:11E20 #msc:11E25 #msc:11E81
paper · pdf · doi:10.1016/j.jalgebra.2014.11.026
published in Journal of Algebra 426, 243-258 (Elsevier BV) · final version, major revisions in the style of writing (abstract and introduction rewritten) compared to v.1
openalex publication_date 2015/01/09 · arxiv created 2015/01/12 · arxiv updated 2015/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.