Sums of three squares of fractions of two variables
type: Item
instance of: scholarly article
title: Sums of three squares of fractions of two variables
zbMATH Open document ID: 1118.11020
author: Olivier Macé
author: Louis Mahé
published in: Manuscripta Mathematica
publication date: 2005-05-13T00:00:00Z
review text: The Pythagoras number of the field \(\mathbb{R}(X,Y)\) is known to be 4 (i.e., every sum of squares is a sum of at most 4 squares, and there are sums of squares that are not sums of 3 squares). Every positive semi-definite polynomial in \(\mathbb{R}[X,Y]\) is a sum of squares in the quotient field, hence is a sum of at most four squares. The authors exhibit several new families of positive semi-definite polynomials in \(\mathbb{R}[X,Y]\) that cannot be written as sums of fewer than 4 squares in \(\mathbb{R}(X,Y)\). Their polynomials are monic of degree 4 in the variable \(Y\), the coefficients are polynomials in the variable \(X\). The coefficients are used to define certain elliptic curves over the field \(\mathbb{R}(X)\). The analysis of these curves yields criteria to decide whether or not a polynomial is a sum of 3 squares.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 2168081
zbMATH Keywords: rational functions
zbMATH Keywords: sums of squares
zbMATH Keywords: Pythagoras number
zbMATH Keywords: elliptic curve
zbMATH Keywords: Jacobian variety
zbMATH Keywords: divisor
MaRDI profile type: Publication
full work available at URL: s00229-004-0535-0
OpenAlex ID: W1603254347
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cites work: Zur Darstellung definiter Funktionen als Summe von Quadraten. (To the representation of definite functions as the sum of squares)
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DOI: 10.1007/S00229-004-0535-0
description: scientific article; zbMATH DE number 2168081