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The Puiseux Characteristic of a Goursat Germ

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Abstract

Germsof Goursat distributions can be classified according to a geometric coding called an RVT code. Jean (ESAIM Control Optim Calc Var. 1:241–266, 1996) and Mormul (Cent Eur J Math 2:859–883, 2004) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii (Mem Amer Math Soc 203(956):x+137, 2010) have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here, we derive a simple formula, Theorem 2, for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector. The simplicity of our theorem (compared with the more complex algorithms previously known) suggests a deeper connection between singularity theory and the theory of nonholonomic distributions.

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Notes

  1. Note that, here, as in [5], we allow r v+1 ≥ 0. In [7], where the letters R, V, and T are replaced by G, S, and T, respectively, the code always begins with two letters G. Thus, the GST codes in [7] are two letters longer than the RVT codes here and in [5].

  2. In [7], the quantity n 1 tells us the position of the last occurring letter R in a critical RVT code. Here, we have n 1 = q − 2, which means r q−1 ≠ 0 and r i = 0 for i < q − 1; in other words, ω is the largest entirely critical string at the tail of (α).

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Acknowledgments

The author would like to thank Richard Montgomery, Wyatt Howard, and Alex Castro for helpful discussion, advice, and most importantly, motivation. The author is also grateful to the referee, whose suggestions greatly improved the overall presentation of this paper.

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  1. Department of Mathematics and Statistics, California State University, Sacramento, Sacramento, CA, USA

    Corey Shanbrom

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  1. Corey Shanbrom
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Correspondence to Corey Shanbrom.

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Shanbrom, C. The Puiseux Characteristic of a Goursat Germ. J Dyn Control Syst 20, 33–46 (2014). https://doi.org/10.1007/s10883-013-9207-2

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