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On solutions of real analytic equations

Author: Tejinder S. Neelon
Journal: Proc. Amer. Math. Soc. 125 (1997), 2531-2535
MSC (1991): Primary 14B12; Secondary 32B99
MathSciNet review: 1396991
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Abstract: Analyticity of ${\c C}^{\infty }$ solutions $y_i =f_i(x), 1\le i\le m$, of systems of real analytic equations $p_j(x,y)= 0, 1\le j\le l$, is studied. Sufficient conditions for ${\c C}^{\infty }$ and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by $p_1,p_2,\ldots ,p_l$. In a special case when the $p_i$'s are independent of $x$, we prove that if a ${\c C}^{\infty }$ solution $h$ satisfies the condition $\det \left( \frac {\partial p_i}{py_j}\right )(h(x)) \not \equiv 0$, then $h$ is necessarily real analytic.

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  • 1. Artin, M., On the Solutions of Analytic Equations. Invent. Math. 5 277-291 (1968). MR 38:344
  • 2. Baouendi, M. S., and Rothschild, L. P., Images of real hypersurfaces under holomorphic mappings. J. Differential Geometry 36 (1992), 75-88. MR 94i:32028
  • 3. Baouendi, M. S., H. Jacobowitz and F. Treves, On analyticity of CR mappings. Ann. Math. 122 (1985), 365-400. MR 87f:32044
  • 4. Baouendi, M. S. and L. P. Rothschild, Germs of CR maps between real analytic hypersurfaces Invent. Math. 93 (1988) no. 3, 481-500. MR 90a:32036
  • 5. Bierstone, E. and P. D. Milman, Semianalytic and Subanalytic sets IHES Publ. Math.67 Paris 1988, 5-42. MR 89k:32011
  • 6. Bochnak, J. Analytic Functions in Banach Spaces. Studia Mathematica, T. XXXV.(1970).
  • 7. Bochnak, J. and J. Siciak Analytic Functions in Topological Vector Spaces. Studia Mathematica, T. XXXIX.(1971). MR 47:2365
  • 8. Malgrange, B., Ideals of differentiable functions. Tata Institute of Fundamental Research, Bombay, Oxford University Press, 1966. MR 35:3446
  • 9. Neelon, T. S.Holomorphic Extensions of CR Functions and CR Mappings Ph. D. thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ 08901. (1993)
  • 10. Siciak, J. A Characterization of Analytic Functions of $n$ Real Variables. Studia Mathematica, T. XXXV.(1970). MR 43:4986
  • 11. Tougeron, J. C., Ideaux de fonctions differnetiables, Springer-Verlag, 1972.

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Additional Information

Tejinder S. Neelon
Affiliation: College of Arts and Sciences, California State University San Marcos, San Marcos, California 92096

Keywords: Power series rings, real analytic equations, semianalytic sets
Received by editor(s): August 15, 1994
Received by editor(s) in revised form: February 2, 1995, October 9, 1995, and March 18, 1996
Communicated by: Eric Bedford
Article copyright: © Copyright 1997 American Mathematical Society

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