Proceedings of the Japan Academy, Series A, Mathematical Sciences

Equisingularity in $R^2$ as Morse stability in infinitesimal calculus

Tzee-Char Kuo and Laurentiu Paunescu

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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).

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 6 (2005), 115-120.

First available in Project Euclid: 2 August 2005

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Primary: 14Pxx: Real algebraic and real analytic geometry

Morse equisingularity infinitesimals Newton Polygon


Kuo, Tzee-Char; Paunescu, Laurentiu. Equisingularity in $R^2$ as Morse stability in infinitesimal calculus. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 6, 115--120. doi:10.3792/pjaa.81.115.

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