Bulletin of the Korean Mathematical Society (대한수학회보)

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A POLAR, THE CLASS AND PLANE JACOBIAN CONJECTURE

  • Joe, Do-Sang (Department of Mathematics Education, Konkuk University)
  • Published : 2010.01.31
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Abstract

Let P be a Jacobian polynomial such as deg P = $deg_y$ P. Suppose the Jacobian polynomial P satisfies the intersection condition satisfying $dim_C$ C[x, y]/$P_y$> = deg P - 1, we can prove that the Keller map which has P as one of coordinate polynomial always has its inverse.

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References

  1. S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, Mathematical Surveys and Monographs, 35. American Mathematical Society, Providence, RI, 1990.
  2. S. S. Abhyankar and T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.
  3. E. Brieskorn and H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag, Basel, 1986.
  4. N. V. Chau, Plane Jacobian polynomial for rational polynomials, preprint, arXiv: 0804.3172v2
  5. D. Cox, J. Little, and D. O'Shea, Using Algebraic Geometry, Second edition. Graduate Texts in Mathematics, 185. Springer, New York, 2005.
  6. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhauser Verlag, Basel, 2000.
  7. A. van den Essen and J. Yu, The D-resultant, singularities and the degree of unfaithfulness, Proc. Amer. Math. Soc. 125 (1997), no. 3, 689–695. https://doi.org/10.1090/S0002-9939-97-03639-3
  8. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser Boston, Inc., Boston, MA, 1994.
  9. R. C. Heitmann, On the Jacobian conjecture, J. Pure Appl. Algebra 64 (1990), no. 1, 35–72. https://doi.org/10.1016/0022-4049(90)90005-3
  10. D. Joe and H. Park, Emdeddings of line in the plane and Abhynkar-Moh epimorphism theorem, Bull. Korean. Math. Soc. 46 (2009), no. 1, 171–182. https://doi.org/10.4134/BKMS.2009.46.1.171
  11. T. Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math. 340 (1983), 140–212.
  12. M. Razar, Polynomial maps with constant Jacobian, Israel J. Math. 32 (1979), no. 2-3, 97–106. https://doi.org/10.1007/BF02764906
  13. L. D. Trang, Simple rational polynomials and the Jacobian conjecture, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 641–659. https://doi.org/10.2977/prims/1210167339