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 Parusinski
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 579-601
Published electronically: February 11, 2004
MathSciNet review: 2047682
Full-text PDF

Abstract | References | Additional Information

Abstract: 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).

References [Enhancements On Off] (What's this?)

  • 1. S. S. Abhyankar and A. Assi, Jacobian of meromorphic curves, Proceedings of the Indian Academy of Sciences (Math. Sci.) 109 No. 2, (May, 1999), 117-163.
  • 2. S. S. Abhyankar and A. Assi, Factoring the Jacobian, Contemporary Mathematics, Vol. 266, 2000.
  • 3. A. Assi, Meromorphic Plane Curves, Math. Z., 230 (1999), 165-183.
  • 4. E. Garcia-Barroso, Sur les courbes polaires d'une courbe plane réduite, Proc. London Math. Soc., (3) 81 (2000), 1-28.
  • 5. E. Garcia-Barroso and B. Teissier, Concentration multi-échelles de courbure dans des fibres de Milnor, Comment. Math. Helv., 74 (1999), 398-418.
  • 6. M. Boguslawska, On the Lojasiewicz exponent of the gradient of holomorphic functions, Bull. Polish Acad. Sci. Math. 47 (1999), no. 4, 337-343.
  • 7. E. Casas-Alvero, Singularities of polar curves, Compositio Math. 89 (1993), 339-359.
  • 8. H. Eggers, Polarinvarianten und die Topologie von Kurvensingulariten, Bonner Math. Schr., 147, (1983).
  • 9. S. Izumi, S. Koike and T.-C. Kuo, Computations and stability of the Fukui invariants, Compositio Math. 130 (2002), 49-73.
  • 10. T.-C. Kuo and Y.C. Lu, On analytic function germs of two complex variables, Topology, 16 (1977), 299-310.
  • 11. T.-C. Kuo and A. Parusinski, Newton Polygon Relative to an Arc, in Real and Complex Singularities (São Carlos, 1998), Chapman & Hall Res. Notes Math., 412, 2000, 76-93.
  • 12. T.-C. Kuo and A. Parusinski, On Puiseux roots of Jacobians, Proc. Japan Acad., 78, Ser. A (2002), 55-59.
  • 13. Lê Dung Tràng, Topological use of polar curves, in Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 507-512. Amer. Math. Soc., Providence, R.I., 1975.
  • 14. M. Merle, Invariants polaires des courbes planes, Invent. Math., 41 (1977), 299-310.
  • 15. F. Pham, Deformations equisingularitiés des ideaux jacobiens de courbes planes, in Proc. Liverpool Singularities Symposium, II, (ed. C.T.C. Wall), (1971), 218-233, Lecture Notes Math., 209, Springer-Verlag.
  • 16. R. J. Walker, Algebraic Curves, Springer-Verlag, (1972).
  • 17. O. Zariski, Studies in equisingularity I, Amer J. Math. 2, 87, (1965), 507-536.

Additional Information

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

Adam Parusinski
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