More about dicriticals

Author:
Shreeram S. Abhyankar

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3083-3097

MSC (2010):
Primary 14A05

Published electronically:
January 24, 2011

MathSciNet review:
2811264

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface.

**[Ab1]**Shreeram Abhyankar,*On the valuations centered in a local domain*, Amer. J. Math.**78**(1956), 321–348. MR**0082477****[Ab2]**Shreeram Abhyankar,*Ramification theoretic methods in algebraic geometry*, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. MR**0105416****[Ab3]**S. S. Abhyankar,*Resolution of singularities of embedded algebraic surfaces*, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1617523****[Ab4]**S. S. Abhyankar,*Lectures on algebra. Vol. I*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR**2253320****[Ab5]**S. S. Abhyankar,*Inversion and Invariance of Characteristic Terms, Part I,*Developments in Mathematics, Springer, 2010.**[Ab6]**Shreeram S. Abhyankar,*Dicritical divisors and Jacobian problem*, Indian J. Pure Appl. Math.**41**(2010), no. 1, 77–97. MR**2650101**, 10.1007/s13226-010-0017-x**[Ab7]**S. S. Abhyankar,*Pillars and Towers of Quadratic Transformations,*Proceedings of the American Mathematical Society, to appear.**[AbH]**S. S. Abhyankar and W. J. Heinzer,*Existence of dicritical divisors,*American Journal of Mathematics, to appear.**[AbL]**S. S. Abhyankar and I. Luengo,*Algebraic theory of dicritical divisors,*American Journal of Mathematics, to appear.**[ZaS]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. MR**0389876**

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Additional Information

**Shreeram S. Abhyankar**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
ram@cs.purdue.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10732-9

Keywords:
Criticals,
special pencil

Received by editor(s):
January 30, 2010

Received by editor(s) in revised form:
August 7, 2010

Published electronically:
January 24, 2011

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.