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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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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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