Nonexistence of C^infinity or formal power series variation in solutions to Hilbert's 17th problem
by
Charles N. Delzell
Louisiana State University

[Note: This abstract supersedes the abstract submitted May 1, 1999.]

Recall Hilbert's 17th problem: If f in R[X₁, ..., X_n] is positive semidefinite (psd), then f=sum p_ir_i², for some nonnegative p_i in R and some r_i in R(X₁, ..., X_n). (Below it will be helpful not to absorb the p_i into the r_i, which one would otherwise do.) Fix d in {0, 1, 2, ... }, and consider the set of all psd f of degree <= d. I have recently proved that for d > 2, there can be no solution to Hilbert's 17th problem in which the p_i and the coefficients of the r_i vary in a C^infinity manner as functions of the coefficients of f; nor can they be taken to be formal power series in the coefficients of f. I had already shown in 1990 (Recent Advances in Real Algebraic Geometry and Quadratic Forms, Contemp. Math., AMS, 1994), that analytic variation is impossible; and Gonzalez-Vega and Lombardi showed (Math. Z., 1997) that for every fixed, finite r >= 0, C^r variation is possible (improving the continuous, piecewise-polynomial variation found in 1988 and 1991 by me and by González-Vega/Lombardi, respectively; the latter improved my continuous, semialgebraic variation of 1980). The impossibility of C^infinity variation follows from the impossibility of formal power series variation, which, in turn, is based on a refinement of the proof of the impossibility of analytic variation.

Date received: June 8, 1999