Impossibility of C∞ variation or formal power series variation in solutions to Hilbert's 17th problem
No matter how a positive semidefinite polynomial f ∈ ℝ[X1,...,Xn] is represented (according to E. Artin's 1926 solution to Hilbert's 17th problem) in the form f = ∑ piri2 (with 0 ≤ Pi ∈ ℝ and ri ∈ ℝ(X1,..., Xn), 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 deg f ≤, 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 r ∈ ℕ, Cr variation is possible, improving upon their and the author's result that continuous, piecewise-polynomial variation is possible. © 2004 Published by Elsevier Inc.
Publication Source (Journal or Book title)
Journal of Algebra
Delzell, C. (2004). Impossibility of C∞ variation or formal power series variation in solutions to Hilbert's 17th problem. Journal of Algebra, 275 (1), 233-249. https://doi.org/10.1016/j.jalgebra.2003.10.032