Impossibility of C∞ variation or formal power series variation in solutions to Hilbertʹs 17th problem

No matter how a positive semidefinite polynomial is represented (according to E. Artinʹs 1926 solution to Hilbertʹs 17th problem) in the form f=∑piri2 (with and ), the pi and the coefficients of the ri cannot be chosen to depend in a C∞ (i.e., infinitely differentiable) manner upon the coefficients of f (unless degf 2); formal powers series variation is also impossible. This answers a question we had raised in 1990 [Contemp. Math., vol. 155, Amer. Math. Soc., 1994, pp. 107–117], where we had already shown that real analytic variation was impossible; and Gonzalez-Vega and Lombardi [Math. Z. 225 (3) (1997) 427–451] then showed that for every fixed, finite , Cr variation is possible, improving upon their and the authorʹs result that continuous, piecewise-polynomial variation is possible.