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 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.[1] A result of Albrecht Pfister[2] shows that a positive semidefinite form in n variables can be expressed as a sum of 2n squares.[3]

Dubois showed in 1967 that the answer is negative in general for ordered fields.[4] In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive coefficients.[5]

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[6] and Procesi, Schacher,[7] with an elementary proof given by Hillar and Nie.[8]

The formulation of the question takes into account that there are polynomials, for example[9]

$f(x,y,z)=z^6+x^4y^2+x^2y^4-3x^2y^2z^2 \,$

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 have been found [10][11] However every real nonnegative polynomial can be approximated as closely as desired (in the $l_1$-norm of its coefficient vector) by a sequence of polynomials that are sums of squares of polynomials.[12]

It is an open question what is the smallest number

$v(n,d), \,$

such that any n-variate, non-negative polynomial of degree d can be written as sum of at most $v(n,d)$ square rational functions over the reals.

The best known result (as of 2008) is

$v(n,d)\leq2^n, \,$

due to Pfister in 1967.[2]

In complex analysis the Hermitian analogue, requiring the squares to be squared norms of holomorphic mappings, is somewhat more complicated, but true for positive polynomials by a result of Quillen.[13] The result of Pfister on the other hand fails in the Hermitian case, that is there is no bound on the number of squares required, see D'Angelo–Lebl.[14]

References

1. ^ Delzell, C.N. (1984). "A continuous, constructive solution to Hilbert's 17th problem". Inventiones Mathematicae 76: 365–384. doi:10.1007/BF01388465. Zbl 0547.12017.
2. ^ a b Pfister, Albrecht (1967). "Zur Darstellung definiter Funktionen als Summe von Quadraten". Inventiones Mathematicae (in German) 4: 229–237. Zbl 0222.10022.
3. ^ Lam (2005) p.391
4. ^ Dubois, D.W. (1967). "Note on Artin's solution of Hilbert's 17th problem". Bull. Am. Math. Soc. 73: 540–541. Zbl 0164.04502.
5. ^ Lorenz (2008) p.16
6. ^ Gondard, Danielle; Ribenboim, Paulo (1974). "Le 17e problème de Hilbert pour les matrices". Bull. Sci. Math. (2) 98 (1): 49–56. MR 432613. Zbl 0298.12104.
7. ^ Procesi, Claudio; Schacher, Murray (1976). "A non-commutative real Nullstellensatz and Hilbert's 17th problem". Ann. of Math. (2) 104 (3): 395–406. MR 432612. Zbl 0347.16010.
8. ^ Nie, Jiawang (2008). "An elementary and constructive solution to Hilbert's 17th problem for matrices". Proc. Am. Math. Soc. 136 (1): 73–76. arXiv:math/0610388. Zbl 1126.12001.
9. ^ Marie-Françoise Roy. The role of Hilbert's problems in real algebraic geometry. Proceedings of the ninth EWM Meeting, Loccum, Germany 1999
10. ^ Lasserre, Jean B. (2007). "Sufficient conditions for a real polynomial to be a sum of squares". Arch. Math. 89 (5): 390–398. Zbl 1149.11018.
11. ^ [1]
12. ^ Lasserre, Jean B. (2007). "A sum of squares approximation of nonnegative polynomials". SIAM Rev. 49 (4): 651–669. ISSN 0036-1445. Zbl 1129.12004.
13. ^ Quillen, Daniel G. (1968). "On the representation of hermitian forms as sums of squares". Invent. Math. 5: 237–242. Zbl 0198.35205.
14. ^ D'Angelo, John P.; Lebl, Jiri (2012). "Pfister's theorem fails in the Hermitian case". Proc. Am. Math. Soc. 140 (4): 1151–1157. arXiv:1010.3215. Zbl 06028329.