Positivstellensatz for semi-algebraic sets in real closed valued fields

Lavi, Noa

doi:10.1090/proc/12595

Positivstellensatz for semi-algebraic sets in real closed valued fields
HTML articles powered by AMS MathViewer

by Noa Lavi PDF

Proc. Amer. Math. Soc. 143 (2015), 4479-4484 Request permission

Abstract:

The purpose of this paper is to give a characterization for polynomials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative Positivstellensatz for sets defined also by valuation terms. For this, we use model theoretic tools, together with existence of canonical valuations.

References

Emil Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 100–115 (German). MR 3069468, DOI 10.1007/BF02952513
Gregory Cherlin and Max A. Dickmann, Real closed rings. II. Model theory, Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231. MR 730855, DOI 10.1016/0168-0072(83)90019-2
M. A. Dickmann, On polynomials over real closed rings, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin-New York, 1980, pp. 117–135. MR 606783
N. Lavi, A Ganzstellensatz for open semi-algebraic sets in real closed valued fields, arXiv:1101.2116v1
Kenneth McKenna, New facts about Hilbert’s seventeenth problem, Model theory and algebra (A memorial tribute to Abraham Robinson), Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975, pp. 220–230. MR 0401720
Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847, DOI 10.1007/BFb0101548
Alexander Prestel and Charles N. Delzell, Positive polynomials, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. From Hilbert’s 17th problem to real algebra. MR 1829790, DOI 10.1007/978-3-662-04648-7
Abraham Robinson, On ordered fields and definite functions, Math. Ann. 130 (1955), 275–271. MR 75932, DOI 10.1007/BF01343896
A. Tarski, A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley 1948.