Impossibility of \(C^{\infty}\) variation or formal power series variation in solutions to Hilbert's 17th problem
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.
scientific article; zbMATH DE number 2105794
scientific article