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Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

Authors
  • Fernando, José F.1
  • 1 Universidad Complutense de Madrid, Madrid, 28040, Spain , Madrid (Spain)
Type
Published Article
Journal
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
Publisher
Springer International Publishing
Publication Date
Jan 09, 2022
Volume
116
Issue
2
Identifiers
DOI: 10.1007/s13398-021-01202-4
Source
Springer Nature
Keywords
Disciplines
  • Original Paper
License
Green

Abstract

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension ≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 3$$\end{document}, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

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