Newton-Puiseux Roots of Jacobian Determinants

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Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=γ(y)$, of the Jacobian $J=f_yg_x-f_xg_y$. We define the tree-model, $T(f,g)$, for the pair, using the contact orders of the Newton-Puiseux roots of $f$ and $g$. We then describe how the $γ$'s climb, and where they leave, the tree. We shall also show by two examples that the way the $γ$'s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one function case where the tree completely determines how the polar roots split away. Our result yield a factorisation of the Jacobian determinant.
19 pages, 2 figures

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Algebraic Geometry, 32S05, 14H20

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http://salesiana.dossiersoluciones.com/handle/123456789/65795

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0211408
 http://arxiv.org/abs/math/0211408

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arXiv