Hilbert's seventeenth problem

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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 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]

See also[edit]

References[edit]

  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. doi:10.1007/bf01425382. 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. doi:10.1090/s0002-9904-1967-11736-1. 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. doi:10.2307/1970962. 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. doi:10.1090/s0002-9939-07-09068-5. 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. doi:10.1007/s00013-007-2251-y. Zbl 1149.11018. 
  11. ^ [1]
  12. ^ Lasserre, Jean B. (2007). "A sum of squares approximation of nonnegative polynomials". SIAM Rev. 49 (4): 651–669. doi:10.1137/070693709. 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. doi:10.1007/bf01389773. 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.