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A continuous, constructive solution to Hilbert’s 17th problem. (English) Zbl 0547.12017

Let K be an ordered field, and K ¯ its unique real closure. A polynomial FK[x 1 ,···x n ] is called positive semi-definite (psd) if f0 in K ¯. Roughly speaking, Artin’s proof shows that f is a sum of squares (SOS) of rational functions r i K(x 1 ,···x n ) provided f is psd and each positive element of K is an SOS. More elegantly, now for all K, f is a weighted SOS: p i r i 2 , where p i K + and f is psd. During the fifties to the dependence of p i r i 2 on f, that is, on its variables x and coefficients c, it was given attention. The best result, by Daykin, has remained unpublished. It provides finitely many representations of f by SOS with terms p i r i 2 such that (i) each p i depends polynomially on c, (ii) each p i r k 2 is rational in both x and c, and (iii) if f is psd then, for one of the representations, all p i 0·

The author gives a new proof of this result, but, in contrast to Daykin, without attention to bounds on the number of terms and their degrees (depending on n and the degree of f). He also treats a natural topological variant where, in place of (ii), p i r i 2 depends rationally on x, and continuously (for the order topology) on both x and c. His principal result achieves this for real closed K, where p i can be absorbed (since |x| is semi-algebraic and continuous), and r i is semi-algebraic in c. Having previously excluded a rational representation without Daykin’s case distinctions, he conjectures that a continuous, piecewise rational solution is possible for all K. Here the ’pieces’ are (basic) semi-algebraic sets. - As the author observed in a later preprint, the last line of p. 366 is false, with a counter-example obtained from the identity on 1.12 of p. 369. However, the false statement is not used in the paper at all, and should be omitted.

Reviewer: G.Kreisel
12J15Ordered fields
11E10Forms over real fields
12D15Formally real fields
11E16General binary quadratic forms
11E76Forms of degree higher than two
03F55Intuitionistic mathematics
03F65Other constructive mathematics
11U99Connections of number theory with logic
54H13Topological fields, rings, etc. (topological aspects)
14PxxReal algebraic and real analytic geometry
[1]Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Sem. Hamburg5, 100-15 (1927). (An English version of the main proof there is on p.289ff. of Jacobson’s Lectures in Abstract Algebra3, 1964) · Zbl 02585749 · doi:10.1007/BF02952513
[2]Bochnak, J., Efroymson, G.: Real algebraic geometry and the 17th Hilbert problem. Math. Ann.251(3), 213-41 (1980) · Zbl 0434.14020 · doi:10.1007/BF01428942
[3]Bourbaki, N.: General Topology, Tome 1. Hermann, Addison-Wesley, Reading, Mass. 1966
[4]Brumfiel, G.: Partially Ordered Rings and Semi-Algebraic Geometry. Lecture Note Series of the London Math. Soc. Cambridge Univ. Press, Cambridge, 1979
[5]Choi, M.D., Lam, T.-Y.: Extremal positive semidefinite forms. Math. Ann.231, 1-18 (1977) · Zbl 0355.10019 · doi:10.1007/BF01360024
[6]Cohen, P.J.: Decision procedures for real andp-adic fields. Comm. in Pure & Applied Math.22, 131-51 (1969) · Zbl 0167.01502 · doi:10.1002/cpa.3160220202
[7]Coste, M.F.: Recursive functions in topoi. Oberwolfach Tagungsberichte 1975
[8]Coste M., Coste-Roy, M.F.: Topologies for real algebraic geometry. Topos Theoretic Methods in Geometry. In: Kock, A. (ed.) Various Publications Series vol. 30. Mathematisk Institut, Aarhus Univ. 1979
[9]Daykin, D. E.: Thesis, Univ. of Reading, 1960 (unpublished); cited by Kreisel, A survey of proof theory. J. Symb. Logic33, 321-88 (1968)
[10]Delzell, C.N.: A constructive, continuous solution to Hilbert’s 17th problem, and other results in semi-algebraic geometry, Ph.D. dissertation, Stanford Univ. 1980 (Univ. Microfilms International, Order No. 8024640). Cf. also Dissertation Abstracts International41, (no. 5) 1980, and AMS Abstracts2(1) (Jan. 1981), # 783-12-28
[11]Delzell, C.N.: Analytic version of Siegel’s theorem on sums of squares, in preparation; preliminary abstract in AMS Abstracts3(2) (Jan. 1982a) #792-12-269
[12]Delzell, C.N.: A finiteness theorem for open semi-algebraic sets, with applications to Hilbert’s 17th problem, Ordered Fields and Real Algebraic Geometry, Dubois, D.W., Recio, T. (eds.). Contemporary Math. Series, AMS, Providence, 1982b, pp. 79-97
[13]Delzell, C.N.: Case distinctions are necessary for representing polynomials as sums of squares. Proc. Herbrand Symp., Logic Coll. 1981, Stern, J. (ed.). North Holland, 1982c, pp. 87-103
[14]Delzell, C.N.: Continuous sums of squares of forms. L.E.J. Brouwer Centenary, Symp. Troelstra, A.S., van Dalen, D. (eds.). North Holland, 1982d, pp. 65-75
[15]Delzell, C.N.: Continuity., rationality, and minimality for sums of squares of linear forms, in preparation; preliminary abstract in AMS Abstracts4 (Jan. 1983) # 801-12-354
[16]Dries, L., van den: Some applications of a model-theoretic fact to (semi-)algebraic geometry. Indag. Math. in press (1984)
[17]Heilbronn, H.: On the representation of a rational as a sum of four, squares by means of regular functions. J. London Math. Soc.39, 72-6 (1964) · Zbl 0131.28903 · doi:10.1112/jlms/s1-39.1.72
[18]Hilbert, D.: Über die Darstellung definiter Formen als Summe von Formenquadraten Math. Ann.32, 342-50 (1888); see also Ges. Abh. vol. 2, pp. 154-61 Berlin-Heidelberg-New York: Springer 1933 · doi:10.1007/BF01443605
[19]Hilbert, D.: Grundlagen der Geometrie (Teubner, 1899); transl. by E.J. Townsend (Open Court Publishing Co. La Salle, IL, 1902); transl. by L. Unger from the tenth German edition (Open Court, 1971)
[20]Hilbert, D.: Mathematische Probleme, Göttinger Nachrichten (1900), pp. 253-97, and Archiv der Mathematik und Physik 3d ser.1, 44-53, 213-37 (1901). Transl. by M.W. Newson, Bull. Amer. Math. Soc.8, 437-79 (1902); reprinted in Mathematical Developments Arising from Hilbert Problems, Browder, F. (ed.), Proc. Symp. in Pure Math.28, Amer. Math. Soc., Providence, 1976, 1-34
[21]Hironaka, H.: Triangulations of semi-algebraic sets. Proc. Symp. in Pure Math.29, Amer. Math. Soc. Providence, 1975, pp. 165-85
[22]Hu, S.-T.: Theory of Retracts. Wayne State Univ. Press, Detroit, 1965
[23]Kreisel, G.: Mathematical significance of consistency proofs. J. Symb. Logic23, 155-82 (1958) (reviewed by A. Robinson, JSL31 128) · Zbl 0088.01502 · doi:10.2307/2964396
[24]Kreisel, G.: Review of Goodstein. Math. Reviews24A, #A1821, 336-7 (1962)
[25]Kreisel, G.: Review of Ershov. Zentralblatt374, 18-9, #02027 (1978)
[26]Kreisel, G., MacIntyre, A.: Constructive logic vs. algebraization L.E.J. Brouwer Cent. Symp. Troelstra, A.S., van Dalen, D. (eds.). North Holland, 1982, pp 217-60
[27]Lam, T.-Y.: The theory of ordered fields. Ring Theory and Algebra. III. McDonald, B. (ed.). New York: Marcel Bekker 1980
[28]McEnerney, J.: Trim stratification of semi-analytic sets. Manuscripta Math.25, (1) 17-46 (1978) · Zbl 0402.32005 · doi:10.1007/BF01170355
[29]Pfister, A.: Hilbert’s 17th problem and related problems on definite forms, Mathematical Developments Arising from Hilbert Problems, Browder, F. (ed.) Proc. Symp. in Pure Math28, 483-489. Amer. Math. Soc. 1976
[30]Recio, T.: Actas de la IV Reunion de Matematicos de Expresion Latina. Mallorca, 1977
[31]Robinson, R.M.: Some definite polynomials which are not sums of squares of real polynomials. Notices Amer. Math. Soc.16, 554 (1969); Selected Questions in Algebra and Logic (Vol. dedicated to the memory of A.I. Mal’cev), Izdat. ?Nauka? Sibirsk Otdel Novosibirsk 264-82 (1973); or Acad. Sci. USSR. (MR49 #2647)
[32]Stengle, G.: A Nullstellensatz and a Positivstellensatz for semi-algebraic geometry. Math. Ann.207, 87-97 (1974) · Zbl 0263.14001 · doi:10.1007/BF01362149
[33]Stengle, G.: Integral solution of Hilbert’s 17th problem. Math. Ann.246, 33-39 (1979) · Zbl 0414.10011 · doi:10.1007/BF01352024
[34]Stout, L.N.: Topological properties of the real numbers object in a topos. Cahiers de Topologie et Géométrie Différentielle17(3), 295-376 (1976)