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June 2005 Equisingularity in $R^2$ as Morse stability in infinitesimal calculus
Tzee-Char Kuo, Laurentiu Paunescu
Proc. Japan Acad. Ser. A Math. Sci. 81(6): 115-120 (June 2005). DOI: 10.3792/pjaa.81.115


Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in $\mathbf{R}^2$: For an analytic family $f_t:(\mathbf{R}^2,0)\to (\mathbf{R},0)$, when should it be called an ``equisingular deformation''? This amounts to finding a suitable trivialization condition (as strong as possible) and, of course, a criterion. The second is on the Morse stability. We define $\mathbf{R}_*$, which is $\mathbf{R}$ ``enriched'' with a class of infinitesimals. How to generalize the Morse Stability Theorem to polynomials over $\mathbf{R}_*$? The space $\mathbf{R}_*$ is much smaller than the space used in Non-standard Analysis. Our infinitesimals are analytic arcs, represented by fractional power series. In our Theorem II, (B) is a trivialization condition which can serve as a definition for equisingular deformation; (A), and (A$'$) in Addendum \refAdd1, are criteria, using the stability of ``critical points'' and the ``complete initial form''; (C) is the Morse stability (Remark 1.6).


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Tzee-Char Kuo. Laurentiu Paunescu. "Equisingularity in $R^2$ as Morse stability in infinitesimal calculus." Proc. Japan Acad. Ser. A Math. Sci. 81 (6) 115 - 120, June 2005.


Published: June 2005
First available in Project Euclid: 2 August 2005

zbMATH: 1096.14044
MathSciNet: MR2159239
Digital Object Identifier: 10.3792/pjaa.81.115

Primary: 14Pxx

Keywords: infinitesimals , Morse equisingularity , Newton polygon

Rights: Copyright © 2005 The Japan Academy


Vol.81 • No. 6 • June 2005
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