Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


A characterization of sums of $ 2n$th powers of global meromorphic functions

Author: Jesús M. Ruiz
Journal: Proc. Amer. Math. Soc. 109 (1990), 915-923
MSC: Primary 32C25; Secondary 12D15
MathSciNet review: 1023347
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a real analytic manifold. In this note we prove

Theorem. Let $ X$ be a compact analytic set of $ M$ and $ \sum $ its singular locus. Then, a meromorphic function $ h$ on $ X$ is a sum of $ 2n$-th powers of meromorphic functions if and only if, for every analytic curve $ \sigma :\left( { - \varepsilon ,\varepsilon } \right) \to X$ not contained in $ \sum $, it holds $ h \circ \sigma = a{t^m} + \cdots $, with $ a > 0$ and $ 2n$ dividing $ m$.

References [Enhancements On Off] (What's this?)

  • [B1] E. Becker, Heredirarily Pythagorean fields and orderings of higher level, IMPA Lecture Notes, no. 29, Rio de Janeiro, 1978. MR 527118 (80f:12021)
  • [B2] -, The real holomorphy ring and sums of $ 2n$-th powers, Lecture Notes in Math., vol. 959, Springer-Verlag, Berlin, Heidelberg, and New York, 1982.
  • [Br-Sch] L. Bröcker and H.-W. Schülting, Valuations of function fields from the geometrical point of view, J. Reine Angew. Math. 365 (1986), 12-32. MR 826150 (87m:12010)
  • [B-W] F. Bruhat and H. Whitney, Quelques propriétés fondamentales des ensembles analytiques réels, Comment. Math. Helv. 33 (1959), 132-160. MR 0102094 (21:889)
  • [C] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77-99. MR 0094830 (20:1339)
  • [EGA] A. Grothendieck and J. Dieudonné, Elements de géométrie algebrique, Publ. Math. I.H.E.S. 24 (1965). MR 0199181 (33:7330)
  • [F] J. Frisch, Points de platitude d'un morphisme d'espaces analytiques complexes, Invent. Math. 4 (1967), 118-138. MR 0222336 (36:5388)
  • [H1] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109-326. MR 0199184 (33:7333)
  • [H2] -, Introduction to real analytic sets and real analytic maps, Quaderno dei gruppi di ricerca del C.N.R. Ist. Mat. "L. Tonelli", Pisa, 1973. MR 0477121 (57:16665)
  • [K] W. Kucharz, Sums of $ 2n$-th powers of real meromorphic functions (to appear).
  • [K-P] F. V. Kuhlmann and A. Prestel, On places of real algebraic function fields, J. Reine Angew. Math. 358 (1984), 181-195. MR 765832 (86d:12014)
  • [M] H. Matsumura, Commutative algebra, 2nd ed., W. A. Benjamin Co., Amsterdam, 1980. MR 575344 (82i:13003)
  • [Rz] J. Ruiz, On Hilbert's 17th problem and real Nullstellensatz for global analytic functions, Math. Z. 190 (1985), 447-454. MR 806902 (87b:32010)
  • [Sch] H-W Schülting, Prime divisors on real varieties and valuation theory, J. Algebra 98 (1986), 499-514. MR 826139 (87f:14010)
  • [T] J. C. Tougeron, Idéaux de fonctions différentiables, Ergeb. Math. 71, Springer-Verlag, Berlin, Heidelberg, and New York, 1972. MR 0440598 (55:13472)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32C25, 12D15

Retrieve articles in all journals with MSC: 32C25, 12D15

Additional Information

PII: S 0002-9939(1990)1023347-0
Article copyright: © Copyright 1990 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia