Some remarks on the computation of Pythagoras numbers of real irreducible algebroid curves through Gram matrices.
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.
scientific article; zbMATH DE number 2066821