Publication: On the positive extension property and Hilbert's 17th problem for real analytic sets.
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2008
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Walter de Gruyter
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In this work we study the Positive Extension (pe) property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X-0 of R-0(n) has the pe property if every positive semidefinite analytic function germ on X-0 has a positive semidefinite analytic extension to R-0(n); analogously one states the pe property for a global real analytic set X in an open set Q of R-0(n). These pe properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ X-0 subset of R-0(3) has the pe property if and only if every positive semidefinite analytic function germ on X-0 is a sum of squares of analytic function germs on X-0; and (2) a global real analytic set X of dimension <= 2 and local embedding dimension <= 3 has the pe property if and only if it is coherent and all its germs have the pe property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X. Moreover, we classify the singularities with the pe property.
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F. Acquistapace, F. Broglia, J.F. Fernando, J.M. Ruiz: On the finiteness of Pythagoras numbers of real meromorphic functions, preprint RAAG 2006. http://www.uni-regensburg.de/Fakultaeten/nat Fak /RAAG/preprints/0185.html
F. Acquistapace, F. Broglia, J.F. Fernando, J.M. Ruiz: On the Pythagoras numbers of real analytic curves, preprint RAAG 2004. http://www.uni-regensburg.de/Fakultaeten/nat Fak /RAAG/preprints/0082.html
F. Acquistapace, F. Broglia, J.F. Fernando, J.M. Ruiz: On the Pythagoras numbers of real analytic surfaces, Ann. Sci. ´ Ecole Norm. Sup. 38, no. 5, 751-772 (2005).
C. Andradas, L. Br¨ocker, J.M. Ruiz: Constructible Sets in Real Geometry, Ergeb. Math. Grenzgeb. 33, Berlin–Heidelber–New York: Springer–Verlag, 1996.
C. Andradas, A. Diaz-Cano, J.M. Ruiz: The Artin–Lang property for normal real analytic surfaces, J. reine angew. Math. 556, 99-111 (2003).
M. Artin: On the solution of analytic equations, Invent. Math. 5, 227-291 (1968).
J. Bochnak, M. Coste, M.F. Roy: Real Algebraic Geometry, Ergeb. Math. 36, Berlin–Heidelberg–New York: Springer–Verlag, 1998.
J. Bochnak, W. Kucharz, M. Shiota: On equivalence of ideals of real global analytic functions and the 17th Hilbert problem, Invent. Math. 63, 403-421 (1981).
F. Broglia, L. Pernazza: An Artin–Lang property for germs of C1 functions. J. Reine Angew. Math. 548, 129-147 (2002).
H. Cartan: Vari´et´es analytiques r´eelles et vari´et´es analytiques complexes. Bull. Soc. Math. France 85, 77-99 (1957).
M.D. Choi, Z.D. Dai, T.Y.Lam, B. Reznick: The Pythagoras number of some affine algebras and local algebras, J. reine Angew. Math. 336, 45-82 (1982).
S. Coen: Sul rango dei fasci coerenti, Boll. Un. Mat. Ital. 22, 373-383 (1967).
J.F. Fernando: On the Pythagoras numbers of real analytic rings, J. Algebra 243, 321-338 (2001).
J.F. Fernando: Positive semidefinite germs in real analytic surfaces, Math. Ann. 322, 49-67 (2002).
J.F. Fernando: Sums of squares in real analytic rings, Trans. Am. Math. Soc. 354, 1909-1919 (2002).
J.F. Fernando: Analytic surface germs with minimal Pythagoras number, Math. Z. 244, no. 4, 725-752 (2003).
J.F. Fernando: On the 17th Hilbert Problem for global analytic functions on dimension 3, preprint RAAG 2005. http://www.uni-regensburg.de/Fakultaeten/nat Fak I/RAAG/preprints/0148.html
J.F. Fernando, J.M. Ruiz: Positive semidefinite germs on the cone, Pacific J. Math. 205, 109-118 (2002).
R. Gunning, H. Rossi: Analytic functions of several complex variables. Englewood Cliff: Prentice Hall, 1965.
J. Harris: Algebraic Geometry, A First Course, Graduate Text in Math. 133, Berlin–Heidelberg–New York: Springer–Verlag, 1992.
P. Jaworski: Positive definite analytic functions and vector bundles. Bull. Polish Ac. Sc. 30, no. 11-12, 501-506 (1982).
P. Jaworski: Extension of orderings on fields of quotients of rings of real analytic functions. Math. Nachr. 125, 329-339 (1986).
T. de Jong, G. Pfister: Local Analytic Geometry, basic theory and applications, Advanced Lectures in Mathematics. Braunschweig/Wiesbaden: Vieweg, 2000.
H. Kurke, T. Mostowski, G. Pfister, D. Popescu, M. Roczen: Die Approximationseigenschaft lokaler Ringe, Lecture Notes in Math. 634, Berlin: Springer– Verlag, 1978.
R. Narasimhan: Introduction to the theory of analytic spaces, Lecture Notes in Math. 25, Berlin–Heidelberg–New York: Springer–Verlag, 1966.
J.M. Ruiz: On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190, 447-459 (1985).
J.M. Ruiz: The basic theory of power series, Advanced Lectures in Mathematics. Braunschweig–Wiesbaden: Vieweg–Verlag, 1993.
J.M. Ruiz: Sums of two squares in analytic rings, Math. Z. C. Scheiderer: On sums of squares in local rings, J. Reine Angew. Math. 540, 205-227 (2001).
A. Tognoli: Propriet`a globali degli spazi analitici reali. Ann. Mat. Pura Appl. (4) 75, 143-218 (1967).
J.C. Tougeron: Id´eaux de fonctions diff´erentiables. Ergeb. Math. 71, Berlin–New York: Springer–Verlag, 1972.