|
Ordered fields satisfying Pólya's theorem
Author(s):
Zeng
Guangxing
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2921-2926.
MSC (2000):
Primary 12J15;
Secondary 12D15
Posted:
April 25, 2005
MathSciNet review:
2159770
Retrieve article in:
PDF
This article is available free of charge
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.
References:
-
- 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:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
