From: reznick@mimas.math.uiuc.edu (Bruce Reznick) Newsgroups: sci.math.research Subject: Re: Positive Polynomials That Are Sums Of Squares Of Polynomials Date: 5 Feb 1996 16:20:31 GMT In article dkeren@world.std.com (Daniel Keren) writes: > >It is known that every positive polynomial over the reals >is the sum of squares of rational functions (this was >Hilbert's 17th problem). > >I am interested in the "size" of the subset of positive >polynomials that are the sum of squares of polynomials, >not rational functions. In some cases, it is equal to the >set of all positive polynomials (for instance, forms with >3 variables and of degree 4). > >Can anything be said, in general, about the "size" of this >subset? For instance, is it dense within the positive >polynomials? > >Thanks, > >-Danny Keren. >-- >Lies written in ink can never disguise facts written in blood. > >-Lu Xun. > Choi and Lam introduced the notation P_{n,m} to represent the set of positive semidefinite forms in n variables with degree m and \Sigma_{n,m} to denote those which are sums of squares of forms. Raphael Robinson proved in 1969 that \Sigma_{n,m} is closed; in fact both these sets are closed convex cones. There exist positive definite forms which are in the interior of P_{n,m} \setminus \Sigma_{n,m}. Some recent surveys of this subject, which include extensive references, are: "Sums of squares of real polynomials" by M. D. Choi, T. Y. Lam and B. Reznick, Proc. Sympos. Pure Math. v 58.2 (1995), 103--126 and three of my papers: "Sums of even powers of real linear forms", Memoirs AMS, no 463 (1992), especially chapters 1,2,3 "Uniform denominators in Hilbert's 17th Problem", Math. Z. 220(1995), 75-97 "Some concrete aspects of Hilbert's 17th Problem", preprint to appear in the proceedings of the Delon-Dickmann-Gondard seminar at Univ. Paris VI. (Available upon request to the author) Bruce Reznick ============================================================================== From: Bruce Reznick Date: Sat, 13 Mar 1999 15:29:33 -0600 (CST) To: rusin@vesuvius.math.niu.edu Subject: Hilbert's 17th problem Dear Dave, The following reference is an "upgrade" for my survey paper "Some Concrete Aspects of Hilbert's 17th Problem", which you referenced in the recent sci.math thread. It is downloadable from: http://www.math.uiuc.edu/Reports/reznick/98-002.html ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Nonnegative polynomials Date: 12 Mar 1999 16:43:09 GMT David G Radcliffe wrote: >Let P(x) be a polynomial with real coefficients. If P(x) >= 0 for all >real x, then there exist polynomials R(x), Q(x) with real coefficients >so that P(x) = Q(x)^2 + R(x)^2 for all real x. Cool - I never thought about that. What we call "the multiplicativity of the norm map" in complex analysis and is used to establish results about sums of two squares in number theory can be applied equally well as long as we know we can factor P into a product of terms (x-a)^2 + b^2. >Does there exist a polynomial function of two real variables such that >P(x,y) >= 0 for all real x,y but P(x,y) is not a sum of squares >of polynomials with real coefficients? Yes; try P(x,y) = 1 + x^2*y^4 + x^4*y^2 - 3*x^2*y^2 . This example due to Motzkin, 1966. Replace "polynomials" by "rational functions" and you have Hilbert's 17th problem, which is how I found this example: there's a nice summary of related problems in A. Pfister's discussion of Hilbert's 17th Problem in the books, "Mathematical Developments Arising from the Hilbert Problems", Proc Symp Pure Math v 28 1976. To summarize the problem: in any ordered ring one may ask whether all positive elements are sums of squares, and if so, for the maximum number r required. If the ring is R, then of course the answer is "yes", and r=1. If the ring is R[x], the answer as Radcliffe suggests is "yes", and r=2. If the ring includes R[x,y], an answer is "no"; Hilbert had a nonconstructive proof. If the ring is R(x) (real rational function of one variable), the answer is again "yes", with r=2. If the ring is R(x,y), then Hilbert proved the answer is "yes" as soon as r >= 4. Artin (1926) proved the answer is "yes" for all fields of rational functions in n variables over any real closed field. A succession of variations and extensions yielded "improved" results; see Pfister's paper for details. I think the nicest of these is Pfister's result that r is at most 2^n . (Note that a priori there need not be a finite value of r, even if every rational function is a sum of finitely many squares.) For the number-theoretic crowd, let me observe that the question is valid for rings such as Z, Q, Q[x], Q(x), and Q(x_1, ..., x_n). I don't know all the conclusions have been drawn, although it is classically true that r=4 for Z and Q. Landau showed r=8 is enough for Q(x), but I don't think that's optimal. As I understand it, the answer is "yes" for the last ring, but it's not known that there is a finite value of r. Of course you could move from squares to some other power; number-theorists recognize this as Waring's problem. It is known that the corresponding number "r" is for these other exponents smaller for Q than for Z. Some further references to Hilbert's 17th problem: index/26CXX.html Waring's problem: index/11PXX.html dave ============================================================================== From: tkeller@nass.usda.gov (Timothy P. Keller) Newsgroups: sci.math Subject: Nonnegative Polynomials Date: 12 Mar 1999 13:35:09 -0500 If P(x,y) > 0 for all ( x,y ) in R^2 then P is a sum of squares of real polynomials. If P(x,y) >= 0 for all (x,y) in R^2, I remember hearing that the conclusion does not necessarily hold---- if I can remember the counter example I will certainly send it to you! tim ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Newsgroups: sci.math Subject: Re: Nonnegative polynomials Date: 12 Mar 1999 01:17:52 -0500 In article <7c9uuc$4d0$1@uwm.edu>, David G Radcliffe wrote: >Let P(x) be a polynomial with real coefficients. If P(x) >= 0 for all >real x, then there exist polynomials R(x), Q(x) with real coefficients >so that P(x) = Q(x)^2 + R(x)^2 for all real x. > >Does there exist a polynomial function of two real variables such that >P(x,y) >= 0 for all real x,y but P(x,y) is not a sum of squares >of polynomials with real coefficients? > >-- >David Radcliffe radcliff@alpha2.csd.uwm.edu Here are some references. Have fun, ZVK(Slavek) A Partial List of Publications of Man-Duen Choi (on Positive linear mappings and sums of squares) 1. ``Positive linear maps on C*-algebras", Canad. J. Math. 24 (1972), pp. 520-529. 2. ``A Schwarz inequality for positive linear maps on C*- algebras", Illinois J. Math. 18(1974), pp. 565-574. 3. ``Completely positive linear maps on complex matrices", Linear Algebra Appl. 10 (1975), pp. 285-290. 4. ``Positive semi-definite biquadratic forms", Linear Algebra Appl. 12(1975), pp. 95-100. 5. with T.Y. Lam, ``An old question of Hilbert", Conference on Quadratic Forms, 1976, edited by G. Orzech, Queen's Paper in Pure and Applied Math. 46(1977), pp. 385-405. 6. with T.Y. Lam, ``Extremal positive semi-definite forms", Math. Ann. 231(1977), pp. 1-18. 7. with T.Y. Lam and B. Reznick, ``Real zeros of positive semi- definite forms", Math. Z. 171(1980), pp. 1-26. 8.``Some assorted inequalities for positive linear maps on C*-algebras", J. Operator Theory, 4(1980), pp. 271-285. 9. with T.Y. Lam, B. Reznick and A. Rosenberg, ``Sums of squares in some integral domains", J. Algebra, 65, (1980), pp. 234-256. 10. ``Positive linear maps", 1980 AMS Summer Institute at Kingston, Proc. Sympos. Pure Math., 38(1982), part 2, pp. 583-590. 11. with M.Knebusch, T.Y.Lam and B.Reznick, ``Transversal zeros of positive semidefinite forms", Proc. at Rennes 1981, Lecture notes in Math., Springer-Verlag, 959(1982), pp. 273-298. 12. with Z.D.Dai, T.Y.Lam and B.Reznick, ``The Pythagoras numbers of some affine algebras and local algebras", J. Reine Angew Math., 336(1982), pp. 45-82. 13. with S.K.Tsui, ``Tracial positive linear maps of C*-algebras", Proc. Amer. Math. Soc., 87(1983), pp. 57-61. 14. with T. Ando, ``Non-linear completely positive maps", Proceedings of Conference on Aspects of Positivity in Functional Analysis, Tubingen, Germany, 1985, Elsevier Science Publishers, B.V. (1986), pp. 3-13. 15. with T.Y. Lam, and B. Reznick, ``Even symmetric sextics", Math. Z., 195(1987), pp. 559-580. 16. with T.Y. Lam and B. Reznick, ``Positive sextics and Schur inequalities", J. Algebra, 141(1991), pp. 36-77. 17. with T.Y. Lam and B. Reznick, ``Sums of squares of real polynomials", Proc. Sympos. Pure Math., 58.2(1995), pp. 103-126. 18. with T.Y. Lam, A. Prestel and B. Reznick, ``Sums of 2m-th powers of rational functions in one variable over real closed fields", Math. Zeit., 221(1996), pp. 93-112. ============================================================================== A bottle of champagne awaits the person who can express (1+t^2)/(2+t^2) as a sum of (2n)th powers of elements of Q(t) for all n. See Victoria Powers,"Hilbert's 17th Problem and the Champagne Problem", American Mathematical Monthly, Vol. 103, No. 10, Dec., 1996, pp. 879-887 Available through JSTOR.org to subscribers.