Hilbert's seventeenth problem [SEP] named after
Hilbert's seventeenth problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original question may be reformulated as:
- Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?
Motivation.
The formulation of the question takes into account that there are non-
denominators in Hilbert's Seventeenth Problem", Math. Z., 220 (1995), 75–98 (MR 96e:11056).
- "Some concrete aspects of Hilbert's 17th problem", in Real Algebraic Geometry and Ordered Structures, (C. N. Delzell, J.J. Madden eds.) Cont. Math., 253 (2000), 251–272 (MR 2001i:11042).
- (With V. Powers, C. Scheiderer and F. Sottile) "A new approach to Hilbert's theorem on ternary quartics", C. R. Math.