|
A strict Positivstellensatz for rings of definable analytic functions
Author:
Andreas Fischer
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1415-1422
MSC (2010):
Primary 03C64, 14P10; Secondary 13J30, 26E10
Posted:
December 31, 2012
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Consider an expansion of the real field in which every unary definable continuous function can be ultimately majorized by a definable analytic function. We prove the strict Positivstellensatz for analytic functions which are definable in such structures. The methods also work for a large class of quasianalytic subrings of the ring of those smooth functions that are definable in a polynomially bounded structure.
- 1.
F.
Acquistapace, C.
Andradas, and F.
Broglia, The strict Positivstellensatz for global analytic
functions and the moment problem for semianalytic sets, Math. Ann.
316 (2000), no. 4, 609–616. MR 1758445
(2001g:14087), http://dx.doi.org/10.1007/s002080050346
- 2.
F.
Acquistapace, C.
Andradas, and F.
Broglia, The Positivstellensatz for definable functions on
o-minimal structures, Illinois J. Math. 46 (2002),
no. 3, 685–693. MR 1951235
(2003k:03053)
- 3.
Jacek
Bochnak, Michel
Coste, and Marie-Françoise
Roy, Real algebraic geometry, Ergebnisse der Mathematik und
ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987
French original; Revised by the authors. MR 1659509
(2000a:14067)
- 4.
M. Coste, An Introduction to O-minimal Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.
- 5.
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
(2002k:13044)
- 6.
Lou
van den Dries, Tame topology and o-minimal structures, London
Mathematical Society Lecture Note Series, vol. 248, Cambridge
University Press, Cambridge, 1998. MR 1633348
(99j:03001)
- 7.
Lou
van den Dries and Chris
Miller, Geometric categories and o-minimal structures, Duke
Math. J. 84 (1996), no. 2, 497–540. MR 1404337
(97i:32008), http://dx.doi.org/10.1215/S0012-7094-96-08416-1
- 8.
A. Fischer, Positivstellensätze for differentiable functions, Positivity. DOI: 10.1007/s11117-010-0077-5
- 9.
J.-L.
Krivine, Anneaux préordonnés, J. Analyse Math.
12 (1964), 307–326 (French). MR 0175937
(31 #213)
- 10.
Chris
Miller, Infinite differentiability in
polynomially bounded o-minimal structures, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2551–2555. MR 1257118
(95j:03069), http://dx.doi.org/10.1090/S0002-9939-1995-1257118-1
- 11.
Chris
Miller, Expansions of dense linear orders with the intermediate
value property, J. Symbolic Logic 66 (2001),
no. 4, 1783–1790. MR 1877021
(2003j:03044), http://dx.doi.org/10.2307/2694974
- 12.
J.-P.
Rolin, P.
Speissegger, and A.
J. Wilkie, Quasianalytic Denjoy-Carleman classes
and o-minimality, J. Amer. Math. Soc.
16 (2003), no. 4,
751–777 (electronic). MR 1992825
(2004g:14065), http://dx.doi.org/10.1090/S0894-0347-03-00427-2
- 13.
Konrad
Schmüdgen, The 𝐾-moment problem for compact
semi-algebraic sets, Math. Ann. 289 (1991),
no. 2, 203–206. MR 1092173
(92b:44011), http://dx.doi.org/10.1007/BF01446568
- 14.
Masahiro
Shiota, Approximation theorems for Nash
mappings and Nash manifolds, Trans. Amer. Math.
Soc. 293 (1986), no. 1, 319–337. MR 814925
(87e:58004), http://dx.doi.org/10.1090/S0002-9947-1986-0814925-3
- 15.
Gilbert
Stengle, A nullstellensatz and a positivstellensatz in
semialgebraic geometry, Math. Ann. 207 (1974),
87–97. MR
0332747 (48 #11073)
- 1.
- F. Acquistapace, C. Andradas, F. Broglia, The strict Positivstellensatz for global analytic functions and the moment problem for semianalytic sets. Math. Ann. 316 (2000), no. 4, 609-616. MR 1758445 (2001g:14087)
- 2.
- F. Acquistapace, C. Andradas, F. Broglia, The Positivstellensatz for definable functions on o-minimal structures. Illinois J. Math. 46 (2002), no. 3, 685-693. MR 1951235 (2003k:03053)
- 3.
- J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36. Springer-Verlag, Berlin, 1998. x+430 pp. MR 1659509 (2000a:14067)
- 4.
- M. Coste, An Introduction to O-minimal Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.
- 5.
- A. Prestel, C. N. Delzell, Positive polynomials. From Hilbert's 17th problem to real algebra. Springer Monographs in Mathematics. Berlin: Springer, 2001. viii+268 pp. MR 1829790 (2002k:13044)
- 6.
- L. van den Dries, Tame Topology and O-minimal Structures. LMS Lecture Notes 248, Cambridge University Press, 1998. MR 1633348 (99j:03001)
- 7.
- L. van den Dries, C. Miller, Geometric categories and o-minimal structures. Duke Math. J. 84, no. 2, 497-540 (1996). MR 1404337 (97i:32008)
- 8.
- A. Fischer, Positivstellensätze for differentiable functions, Positivity. DOI: 10.1007/s11117-010-0077-5
- 9.
- J.-L. Krivine, Anneaux préordonnés, J. Analyse Math. 12, 307-326 (1964). MR 0175937 (31:213)
- 10.
- C. Miller, Infinite differentiability in polynomially bounded o-minimal structures. Proc. Amer. Math. Soc. 123, no. 8, 2551-2555 (1995). MR 1257118 (95j:03069)
- 11.
- C. Miller, Expansions of dense linear orders with the intermediate value property. J. Symbolic Logic 66 (2001), no. 4, 1783-1790. MR 1877021 (2003j:03044)
- 12.
- J.-P. Rolin, P. Speissegger, A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751-777. MR 1992825 (2004g:14065)
- 13.
- K. Schmüdgen, The
 -moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203-206. MR 1092173 (92b:44011)
- 14.
- M. Shiota, Approximation theorems for Nash mappings and Nash manifolds. Trans. Amer. Math. Soc. 293, no. 1, 319-337 (1986). MR 814925 (87e:58004)
- 15.
- G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87-97. MR 0332747 (48:11073)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
03C64,
14P10,
13J30,
26E10
Retrieve articles in all journals
with MSC (2010):
03C64,
14P10,
13J30,
26E10
Additional Information
Andreas Fischer
Affiliation:
Gymnasium St. Ursula, Ursulastrasse 8-10, Dorsten, Germany
Address at time of publication:
Comenius Gymnasium Datteln, Südring 150, 45711 Datteln, Germany
Email:
el.fischerandreas@live.de
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11361-9
PII:
S 0002-9939(2012)11361-9
Keywords:
Positivstellensatz,
analytic and smooth function,
polynomially bounded structure
Received by editor(s):
April 21, 2009
Received by editor(s) in revised form:
May 1, 2010, February 7, 2011, and July 12, 2011
Posted:
December 31, 2012
Additional Notes:
The author is a postdoctoral fellow of the Thematic Program on o-minimal Structures and Real Analytic Geometry at the Fields Institute
Communicated by:
Julia Knight
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
