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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
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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$.


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  • [B1] Eberhard Becker, Hereditarily-Pythagorean fields and orderings of higher level, Monografías de Matemática [Mathematical Monographs], vol. 29, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978. MR 527118
  • [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] Ludwig Bröcker and Heinz-Werner Schülting, Valuations of function fields from the geometrical point of view, J. Reine Angew. Math. 365 (1986), 12–32. MR 826150
  • [B-W] H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33 (1959), 132–160 (French). MR 0102094
  • [C] Henri Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77–99 (French). MR 0094830
  • [EGA] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181
  • [F] Jacques Frisch, Points de platitude d’un morphisme d’espaces analytiques complexes, Invent. Math. 4 (1967), 118–138 (French). MR 0222336
  • [H1] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184
  • [H2] Heisuke Hironaka, Introduction to real-analytic sets and real-analytic maps, Istituto Matematico “L. Tonelli” dell’Università di Pisa, Pisa, 1973. Quaderni dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche. MR 0477121
  • [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 algebraic function fields, J. Reine Angew. Math. 353 (1984), 181–195. MR 765832, 10.1515/crll.1984.353.181
  • [M] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • [Rz] Jesús M. Ruiz, On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions, Math. Z. 190 (1985), no. 3, 447–454. MR 806902, 10.1007/BF01215144
  • [Sch] H.-W. Schülting, Prime divisors on real varieties and valuation theory, J. Algebra 98 (1986), no. 2, 499–514. MR 826139, 10.1016/0021-8693(86)90009-8
  • [T] Jean-Claude Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, Berlin-New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. MR 0440598

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DOI: https://doi.org/10.1090/S0002-9939-1990-1023347-0
Article copyright: © Copyright 1990 American Mathematical Society