Impossibility of \(C^{\infty}\) variation or formal power series variation in solutions to Hilbert's 17th problem
type: Item
instance of: scholarly article
title: Impossibility of \(C^{\infty}\) variation or formal power series variation in solutions to Hilbert's 17th problem
zbMATH Open document ID: 1092.14069
author: Charles N. Delzell
published in: Journal of Algebra
publication date: 2004-10-04T00:00:00Z
review text: In 1926 Artin solved Hilbert's 17th problem showing that every positive semi-definite polynomial \(f\in\mathbb{R}[X_1,\dots,X_n]\) can be written in the form \(\sum p_i\cdot g_i^2\), where \(0<p_i\in\mathbb{R}\) and \(g_i\in\mathbb{R}(X_1,\dots,X_n)\). The present paper is concerned with the question how the numbers \(p_i\) and the coefficients of the \(g_i\) depend upon the coefficients of the input polynomial \(f\). It is known that, in general, analytic dependence is impossible, whereas it is possible to choose the \(p_i\) and the \(g_i\) in such a way that, for any fixed finite \(k\), their coefficients can be expressed as \(C^k\)-functions of the coefficients of \(f\). In the present paper it is first shown that formal power series variation cannot be achieved, in general. It is then shown that there are no \(C^\infty\)-functions that provide a representation of the polynomial \(X^4+c_1\cdot X^2+c_2\) in the vicinity of the origin of the coefficient space. The method of proof is to consider Taylor series and thus reduce the problem to formal power series variation.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 12J15
zbMATH DE Number: 2105794
zbMATH Keywords: \(C^\infty\) functions
zbMATH Keywords: sums of squares
zbMATH Keywords: basic closed semianalytic sets
zbMATH Keywords: positive semidefinite polynomials
zbMATH Keywords: Hilbert's 17th problem
zbMATH Keywords: formal power series
zbMATH Keywords: Weierstrass polynomials
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.10.032
OpenAlex ID: W2128916519
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DOI: 10.1016/J.JALGEBRA.2003.10.032
description: scientific article; zbMATH DE number 2105794