Proceedings of the Japan Academy, Series A, Mathematical Sciences

On Puiseux roots of Jacobians

Tzee-Char Kuo and Adam Parusi\'{n}ski

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Take holomorphic $f(x,y)$, $g(x,y)$. A polar arc is a Puiseux root, $x = \gamma(y)$, of the Jacobian $J = f_y g_x - f_x g_y$, but not one of $f \cdot g$. We define the tree, $T(f,g)$, using the contact orders of the roots of $f \cdot g$, describe how polar arcs climb, and leave, the tree, and how to factor $J$ in $\mathbf{C}\{x,y\}$. When collinear points/bars exist, the way the $\gamma$'s leave the tree is not an invariant.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 5 (2002), 55-59.

First available in Project Euclid: 23 May 2006

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Zentralblatt MATH identifier

Primary: 32S05: Local singularities [See also 14J17] 14H20: Singularities, local rings [See also 13Hxx, 14B05]

Puiseux roots polar arcs Jacobian


Kuo, Tzee-Char; Parusi\'{n}ski, Adam. On Puiseux roots of Jacobians. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 5, 55--59. doi:10.3792/pjaa.78.55.

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