Some remarks on the computation of Pythagoras numbers of real irreducible algebroid curves through Gram matrices.
type: Item
instance of: scholarly article
title: Some remarks on the computation of Pythagoras numbers of real irreducible algebroid curves through Gram matrices.
zbMATH Open document ID: 1068.14031
author: José F. Fernando
author: Ronan Quarez
published in: Journal of Algebra
publication date: 2004-05-27T00:00:00Z
review text: In a previous paper [\textit{R. Quarez}, J. Algebra, 238, No. 1, 139--158 (2001; Zbl 1044.14031)] the second author obtained an algorithm to study lengths of sums of squares in the ring \(A\) of a real algebroid curve. As a corollary, the Pythagoras number of \(A\) is upperly bounded by its multiplicity. The proof involves the solvability of a system of linear equations and the case with infinitely many solutions is reduced to the one with a unique solution by means of ``Efroymson trick'', that is, by adding an infinitesimal \(\varepsilon\) to some coefficients. Of course, the argument finishes with \(\varepsilon\to 0\), and at this point some difficulty occurs in the original paper by Quarez. The authors of the short note under review avoid such difficulty with elementary methods and a very clever idea.
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14P99
zbMATH DE Number: 2066821
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.07.023
OpenAlex ID: W1984365318
cites work: The Pythagoras number of some affine algebras and local algebras.
cites work: Pythagoras numbers of real irreducible algebroid curves and Gram matrices
DOI: 10.1016/J.JALGEBRA.2003.07.023
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description: scientific article; zbMATH DE number 2066821