Inversion and Invariance of Characteristic Terms: Part I
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 phrasesInvariance Valuations
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.
- 1.S. K. Abhyankar, Intermediate Algebra, First Edition, Indian Press, Allahabad, 1943; Reprinted in 1960 by Karnatak Printing Press, BombayGoogle Scholar
- 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
- 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.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.S. S. Abhyankar, Lectures on Algebra I, World Scientific, Singapore, 2006Google Scholar
- 20.G. Chrystal, Textbook of Algebra I and II, Cambridge University Press, Cambridge, 1886Google Scholar
- 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
- 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
- 25.C. J. Jan, On polynomial generators of k(x,y), Ph.D. Thesis, Purdue University, 1974Google Scholar
- 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
- 35.O. Zariski, Algebraic Surfaces, Springer, Berlin, 1934Google Scholar