Inversion and Invariance of Characteristic Terms: Part I

Chapter

Summary

In my 1967 paper with almost the same title which appeared in volume 89 of the American Journal of Mathematics, I proved the invariance of the characteristic terms in the fractional power series expansion of a branch of an algebraic plane curve over fields of characteristic zero. Now I extend the results by a more generous interpretation of the characteristic terms, and by relaxing the characteristic zero hypothesis.

Key words and phrases

Invariance Valuations 

Notes

Acknowledgments

On the algebraic side, my thanks are to Pierrette Cassou-Noguès, Bill Heinzer, Giulio Peruginelli, Avinash Sathaye, and Dave Shannon for numerous useful discussions. On the topological side, my thanks are to Dung Trang Le, Walter Neumann, Stepan Orevkov, and Claude Weber for many stimulating discussions. But above all, several private lectures which were given to me by Enrique Artal-Bartolo and Arnaud Bodin in Lille in July 2008 have been most helpful for clarifying the theory of dicritical divisors.

References

  1. 1.
    S. K. Abhyankar, Intermediate Algebra, First Edition, Indian Press, Allahabad, 1943; Reprinted in 1960 by Karnatak Printing Press, BombayGoogle Scholar
  2. 2.
    S. S. Abhyankar, On the valuations centered in a local domain, American Journal of Mathematics, 78 (1956), 321–348MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    S. S. Abhyankar, Ramification Theoretic Methods in Algebraic Geometry, Princeton University Press, Princeton, 1959MATHGoogle Scholar
  4. 4.
    S. S. Abhyankar, Inversion and invariance of characteristic pairs, American Journal of Mathematics, 89 (1967), 363–372MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, American Mathematical Monthly, 83 (1976), 409–448MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    S. S. Abhyankar, Expansion Techniques in Algebraic Geometry, Tata Institute of Fundamental Research, Bombay, 1977MATHGoogle Scholar
  7. 7.
    S. S. Abhyankar, On the semigroup of a meromorphic curve, Part I, Proceedings of the International Symposium on Algebraic Geometry, Kyoto (1977), pp.240–414Google Scholar
  8. 8.
    S. S. Abhyankar, Quasirational singularities, American Journal of Mathematics, 101 (1979), 276–300MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, 1990MATHGoogle Scholar
  10. 10.
    S. S. Abhyankar, Some remarks on the Jacobian question, Purdue Lecture Notes, pp.1–20 (1971); Published in the Proceedings of the Indian Academy of Sciences, 104 (1994), 515–542Google Scholar
  11. 11.
    S. S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces, First Edition of 1966 Published by Academic, New York, Second Enlarges Edition of 1998 Published by Springer, BerlinGoogle Scholar
  12. 12.
    S. S. Abhyankar, Lectures on Algebra I, World Scientific, Singapore, 2006Google Scholar
  13. 13.
    S. S. Abhyankar, Some thoughts on the Jacobian conjecture, Part I, Journal of Algebra, 319 (2008), 493–548MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    S. S. Abhyankar, Some thoughts on the Jacobian conjecture, Part II, Journal of Algebra, 319 (2008), 1154–1248MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    S. S. Abhyankar, Some thoughts on the Jacobian conjecture, Part III, Journal of Algebra, 319 (2008), 2720–2826CrossRefGoogle Scholar
  16. 16.
    E. Artal-Bartolo, Une démonstration géométrique du théorèm d’Abhyankar-Moh, Crelle Journal, 464 (1995), 97–108MathSciNetMATHGoogle Scholar
  17. 17.
    E. T. Bell, Men of Mathematics, Simon & Schuster, New York, 1937MATHGoogle Scholar
  18. 18.
    P. Cassou-Noguès, The effect of rational maps on polynomial maps, Annales Polonici Mathematici, 76 (2001), 21–31MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    P. Cassou-Noguès, Bad field generators, Contemporary Mathematics, 369 (2005), 77–83CrossRefGoogle Scholar
  20. 20.
    G. Chrystal, Textbook of Algebra I and II, Cambridge University Press, Cambridge, 1886Google Scholar
  21. 21.
    D. Eisenbud and W. D. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, 101 (1985)Google Scholar
  22. 22.
    L. Fourrier, Topologie d’un polynome de deux variables complexes au voisinage de l’infini, Annales de l’institut Fourier, 46 (1996), 645–687MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    G. Halphen, Études sur les points singuliers des courbes algébriques planes, Appendix to the French Edition of Salmon’s Higher Plane Curves, 1894, pp.535–648Google Scholar
  24. 24.
    E. W. Hobson, Plane Trigonometry, Cambridge University Press, Cambridge, 1891MATHGoogle Scholar
  25. 25.
    C. J. Jan, On polynomial generators of k(x,y), Ph.D. Thesis, Purdue University, 1974Google Scholar
  26. 26.
    D. T. Le and C. Weber A geometrical approach to the Jacobian conjecture for n = 2, Kodai Journal of Mathematics, 17 (1994), 375–381Google Scholar
  27. 27.
    J. F. Mattei and R. Moussu, Holonomie et intégrales premières, Annales scientifiques de l’cole Normale Suprieure, 13 (1980), 469–523MathSciNetMATHGoogle Scholar
  28. 28.
    M. Nagata, Local Rings, Wiley, New York, 1962MATHGoogle Scholar
  29. 29.
    W. D. Neumann, Complex algebraic plane curves via their links at infinity, Inventiones Mathematicae, 98 (1989), 445–489MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    W. D. Neumann and P. Norbury, Rational polynomials of simple type, Pacific Journal of Mathematics, 204 (2002), 177–207MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    L. Rudolph, Embeddings of the line in the plane, Crelle Journal, 337 (1982), 113–118MathSciNetMATHGoogle Scholar
  32. 32.
    P. Russell, Field generators in two variables, Journal of Mathematics of Kyoto University, 15 (1975), 555–571MathSciNetMATHGoogle Scholar
  33. 33.
    P. Russell, Good and bad field generators, Journal of Mathematics of Kyoto University, 17 (1977), 319–331MathSciNetMATHGoogle Scholar
  34. 34.
    H. J. S. Smith, On the higher singularities of plane curves, Proceedings of the London Mathematical Society, 6 (1873), 153–182CrossRefGoogle Scholar
  35. 35.
    O. Zariski, Algebraic Surfaces, Springer, Berlin, 1934Google Scholar

Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentPurdue UniversityWest LafayetteUSA

Personalised recommendations