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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Newton-Puiseux roots of Jacobian determinants

Authors: Tzee-Char Kuo and Adam Parusiński
Journal: J. Algebraic Geom. 13 (2004), 579-601
Published electronically: February 11, 2004
MathSciNet review: 2047682
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Abstract | References | Additional Information


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 polar root is a Newton-Puiseux root, $x=\gamma (y)$, of the Jacobian $J=f_yg_x-f_xg_y$, but not a root of $f\cdot g$.

We define the tree-model, $T(f,g)$, for the pair, using the set of contact orders of the Newton-Puiseux roots of $f$ and $g$. Our main results (§2) describe how the $\gamma$’s climb, and leave, the tree (like vines). We also show by two examples (§5) that when the tree has what we call collinear points or bars, the way the $\gamma$’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 $T(f)$ completely determines how the polar roots split away.

Our results yield a factorisation of the Jacobian determinant in $\mathbb {C} \{x,y\}$ (§6). As in the one-function case, the factors need not be invariants, nor irreducible. However, some factors do yield invariant truncations and intersection multiplicities (§7).

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Additional Information

Tzee-Char Kuo
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, 2006, Australia

Adam Parusiński
Affiliation: Département de Mathématiques, U.M.R. 6093 du C.N.R.S, Université d’Angers, 2, bd Lavoisier, 49045 Angers Cedex, France

Received by editor(s): April 1, 2002
Published electronically: February 11, 2004
Additional Notes: The first author is partially supported by an ARC Large Grant