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 definite rational functions as sums of quotients of squares. The original question may be stated 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?
This was solved in the affirmative, in 1927, by Emil Artin, for positive definite functions over the reals or more generally real-closed fields. An algorithmic solution was found by Charles Delzell in 1984. A result of Albrecht Pfister shows that a positive semidefinite form in n variables can be expressed as a sum of 2n squares.
Dubois showed in 1967 that the answer is negative in general for ordered fields. In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive coefficients.
A generalization to the matrix case (matrices with rational function entries that are always positive semidefinite are sums of symmetric squares) was given by Gondard, Ribenboim and Procesi, Schacher, with an elementary proof given by Hillar and Nie.
The formulation of the question takes into account that there are polynomials, for example
which are non-negative over reals and yet which cannot be represented as a sum of squares of other polynomials, as Hilbert had shown in 1888 but without giving an example: the first explicit example was found by Motzkin in 1966.
Explicit sufficient conditions for a polynomial to be a sum of squares of other polynomials were found However every real nonnegative polynomial can be approximated as closely as desired (in the -norm of its coefficient vector) by a sequence of polynomials that are sums of squares of polynomials.
It is an open question what is the smallest number
- ,
such that any n-variate, non-negative polynomial of degree d can be written as sum of at most square rational functions over the reals.
The best known result (as of 2008) is
due to Pfister in 1967.
Famous quotes containing the words seventeenth and/or problem:
“The general feeling was, and for a long time remained, that one had several children in order to keep just a few. As late as the seventeenth century . . . people could not allow themselves to become too attached to something that was regarded as a probable loss. This is the reason for certain remarks which shock our present-day sensibility, such as Montaigne’s observation, “I have lost two or three children in their infancy, not without regret, but without great sorrow.””
—Philippe Ariés (20th century)
“What happened at Hiroshima was not only that a scientific breakthrough ... had occurred and that a great part of the population of a city had been burned to death, but that the problem of the relation of the triumphs of modern science to the human purposes of man had been explicitly defined.”
—Archibald MacLeish (1892–1982)