Continuous Pythagoras numbers for rational quadratic forms
type: Item
instance of: scholarly article
title: Continuous Pythagoras numbers for rational quadratic forms
zbMATH Open document ID: 0626.10019
DOI: 10.1016/0022-314X(87)90083-7
author: Charles N. Delzell
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: In a commutative ring R consider the set \(\Sigma\) of elements representable as \(\sum w_ is^ 2_ i\), where \(w_ i\), resp. \(s_ i\), are chosen from subsets W, resp. S, of R. Concrete examples studied in the paper are \(R={\mathbb{R}}[X_ 1,...,X_ n]\), \(W={\mathbb{R}}^+\) or \(W=\{1\}\) and \(S=\{linear\) forms\(\}\). The question is, are there a fixed number m and functions observing special restrictions \(f_ 1,...,f_ m\), resp. \(g_ 1,...,g_ m\), from \(\Sigma\) to W, resp. S, such that for all \(a\in \Sigma\) one has \(a=\sum^{m}_{i=1}f_ i(a)g_ i(a)^ 2 ?\) The minimal possible m is then some sort of Pythagoras number. The restrictions posed on the \(f_ i\), \(g_ i\) are first continuity (of \(f_ i\), \(g_ i\) separately or of \(f_ ig^ 2_ i)\) and secondly some kinds of ''rationality'': \({\mathbb{Q}}\)-rational; semialgebraic, defined by polynomials over \({\mathbb{Q}}\); f(\({\mathbb{Q}})\subset {\mathbb{Q}}\). The author obtains several values or bounds for his Pythagoras numbers.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 12D99
Mathematics Subject Classification ID: 13J99
zbMATH DE Number: 4016998
zbMATH Keywords: sum of squares
zbMATH Keywords: rational quadratic forms
zbMATH Keywords: commutative ring
zbMATH Keywords: Pythagoras numbers
reviewed by: Friedrich Ischebeck
MaRDI profile type: Publication
cites work: On sums of squares and on elliptic curves over function fields
cites work: The Pythagoras number of some affine algebras and local algebras.
cites work: A continuous, constructive solution to Hilbert's \(17^{th}\) problem
cites work: Analytic Right-Inverses to Quadratic Forms over Number Fields
cites work: On the Representation of a Rational as a Sum of Four Squares By Means of Regular Functions
cites work: A Hasse principle for two dimensional global fields.
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description: scientific article; zbMATH DE number 4016998