|
Ordered fields satisfying Pólya's theorem
Author:
Zeng Guangxing
Journal:
Proc. Amer. Math. Soc. 133 (2005), 2921-2926
MSC (2000):
Primary 12J15; Secondary 12D15
Posted:
April 25, 2005
MathSciNet review:
2159770
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The purpose of this paper is to characterize ordered fields satisfying Pólya's theorem on positive representations of polynomials. As a main result, it is proved that an ordered field  satisfies Pólya's theorem if and only if  is an archimedean ordering and  is a real closed field.
- 1.
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)
- 2.
W.
Habicht, Über die Zerlegung strikte definiter Formen in
Quadrate, Comment. Math. Helv. 12 (1940),
317–322 (German). MR 0002837
(2,119f)
- 3.
G.
H. Hardy, J.
E. Littlewood, and G.
Pólya, Inequalities, Cambridge, at the University
Press, 1952. 2d ed. MR 0046395
(13,727e)
- 4.
Nathan
Jacobson, Lectures in abstract algebra. III, Springer-Verlag,
New York, 1975. Theory of fields and Galois theory; Second corrected
printing; Graduate Texts in Mathematics, No. 32. MR 0392906
(52 #13719)
- 5.
E.
de Klerk and D.
V. Pasechnik, Approximation of the stability number of a graph via
copositive programming, SIAM J. Optim. 12 (2002),
no. 4, 875–892 (electronic). MR 1922500
(2003g:90053), http://dx.doi.org/10.1137/S1052623401383248
- 6.
T.
Y. Lam, The theory of ordered fields, Ring theory and algebra,
III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979), Lecture
Notes in Pure and Appl. Math., vol. 55, Dekker, New York, 1980,
pp. 1–152. MR 584611
(82e:12033)
- 7.
Jesús
A. de Loera and Francisco
Santos, An effective version of Pólya’s theorem on
positive definite forms, J. Pure Appl. Algebra 108
(1996), no. 3, 231–240. MR 1384003
(97b:12001), http://dx.doi.org/10.1016/0022-4049(95)00042-9
Jesús
A. de Loera and Francisco
Santos, Erratum to: “An effective version of
Pólya’s theorem on positive definite forms” [J. Pure
Appl. Algebra 108 (1996), no. 3, 231–240; MR1384003
(97b:12001)], J. Pure Appl. Algebra 155 (2001),
no. 2-3, 309–310. MR 1801421
(2001m:11058), http://dx.doi.org/10.1016/S0022-4049(00)00127-4
- 8.
G. Pólya, Über positive Darstellung von Polynomen, Vierteljahrsschrift Natur. Ges. in Zürich, 73(1928), 141-145.
- 9.
Victoria
Powers and Bruce
Reznick, A new bound for Pólya’s theorem with
applications to polynomials positive on polyhedra, J. Pure Appl.
Algebra 164 (2001), no. 1-2, 221–229. Effective
methods in algebraic geometry (Bath, 2000). MR 1854339
(2002g:14087), http://dx.doi.org/10.1016/S0022-4049(00)00155-9
- 10.
Alexander
Prestel, Lectures on formally real fields, Lecture Notes in
Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847
(86h:12013)
- 11.
Abraham
Robinson, Algorithms in algebra, Model theory and algebra (A
memorial tribute to Abraham Robinson), Springer, Berlin, 1975,
pp. 14–40. Lecture Notes in Math., Vol. 498. MR 0401471
(53 #5298)
- 12.
Markus
Schweighofer, An algorithmic approach to Schmüdgen’s
Positivstellensatz, J. Pure Appl. Algebra 166 (2002),
no. 3, 307–319. MR 1870623
(2002j:14063), http://dx.doi.org/10.1016/S0022-4049(01)00041-X
- 1.
- J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Springer-Verlag, New York-Berlin, Heidelberg, 1998. MR 1659509 (2000a:14067)
- 2.
- W. Habicht, Über die Zerlegung strikte definiter Formen in Quadrate, Comm. Math. Helv., 12(1940), 317-322 MR 0002837 (2:119f)
- 3.
- G. H. Hardy, J. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 2nd ed., 1952. MR 0046395 (13:727e)
- 4.
- N. Jacobson, Lectures in Abstract Algebra III, Theory of Fields and Galois Theory, Second corrected printing, Graduate Texts in Math. 32, Springer-Verlag, Berlin, Heidelberg, New York, 1975. MR 0392906 (52:13719)
- 5.
- E. de Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optim., 12(2002), 875-892. MR 1922500 (2003g:90053)
- 6.
- T. Y. Lam, The Theory of Ordered Fields, Lecture Notes in Pure and Appl. Math. 55, Marcel Dekker, New York, 1980. MR 0584611 (82e:12033)
- 7.
- J. A. de Loera and F. Santos, An effective version of Pólya's theorem on positive definite forms, J. Pure Appl. Algebra, 108(1996), 231-240; erratum ibid. 155(2001), 309-310. MR 1384003 (97b:12001); MR 1801421 (2001m:11058)
- 8.
- G. Pólya, Über positive Darstellung von Polynomen, Vierteljahrsschrift Natur. Ges. in Zürich, 73(1928), 141-145.
- 9.
- V. Powers and B. Reznick, A new bound for Pólya's theorem with applications to polynomials positive on polyhedra, J. Pure Appl. Algebra, 164(2001), 221-229. MR 1854339 (2002g:14087)
- 10.
- A. Prestel, Lectures on Formally Real Fields, Lecture Notes in Math. 1093, Springer-Verlag, Berlin, Heidelberg, New York, 1984. MR 0769847 (86h:12013)
- 11.
- A. Robinson, Algorithms in algebra, Model Theory Algebra, Mem. Tribute to Abraham Robinson, Lecture Notes in Math. 498, Springer-Verlag, Berlin, 1975, 14-40. MR 0401471 (53:5298)
- 12.
- M. Schweighofer, An algorithmic approach to Schmüdgen's Positivstellensatz, J. Pure Appl. Algebra, 166(2002), 307-319. MR 1870623 (2002j:14063)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
12J15,
12D15
Retrieve articles in all journals
with MSC (2000):
12J15,
12D15
Additional Information
Zeng Guangxing
Affiliation:
Department of Mathematics, Nanchang University, Jiangxi Province, Nanchang 330047, People’s Republic of China
Email:
zenggx@ncu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002-9939-05-07856-1
PII:
S 0002-9939(05)07856-1
Keywords:
Ordered field,
P\'olya's theorem,
archimedean ordering,
real closed field
Received by editor(s):
March 3, 2004
Received by editor(s) in revised form:
June 10, 2004
Posted:
April 25, 2005
Additional Notes:
This work was partially supported by a National Key Basic Research Project of China (Grant No. 2004CB318003).
Communicated by:
Lance W. Small
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
