**Date & Time:** Tuesday, January 06, 2009, 16:00-17:00.

**Venue:** Ramanujan Hall

**Title:** Inversion and Invariance of Characteristic Terms

**Speaker:** Shreeram S. Abhyankar, Purdue University

**Abstract:** A branch of an algebraic or analytic plane curve can be parametrized by expressing both the variables as power series in a parameter; we call this the MT (= Maclaurin-Taylor) expansion. In case of zero characteristic, by Hensel's Lemma or by Newton's Theorem on fractional power series expansion, one of the variables can be arranged to be a power of the parameter, and then certain divisibility properties of the exponents in the expansion of the other variable lead to the characteristic terms whose importance was first pointed out by Smith (1873) and Halphen (1894) as noted in Zariski's famous book Algebraic Surfaces (1934).

In my 1967 paper with almost the same title as that of this talk, 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.