Impossibility of C ∞ variation or formal power series variation in solutions to Hilbert ’ s 17 th 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 = ∑ pir 2 i (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, AMS, 1994, pp. 107–17), where we had already shown that real analytic variation was impossible; and Gonzalez-Vega and Lombardi (Math. Z. 225(3) (1997), 427–51) 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.