Mathematics Colloquium, Spring 2005

The Nevanlinna-Cartan Second Main Theorem and Functional Equations f₀ⁿ + f₁ⁿ + · · · + f_kⁿ = 0

The purpose of this talk is to give a survey of the value distribution theory of meromorphic functions on (C) and holomorphic curves into the projective space P(C) from the viewpoint of the functional equation

(*)                                f₀ⁿ + f₁ⁿ + · · · + f_kⁿ = 0

R. Nevanlinna's value distribution theory is an effective method of investigating the non-existence of non-trivial meromorphic solutions of the Fermat equation f₁ⁿ + f₂ⁿ + f₃ⁿ = 0. In the 1960's, F. Gross and I.N. Baker proved that there are no such solutions for n > 4, and the Weierstrass -function gives its 'unique' meromorphic solutions for n = 3. Later, W.K. Hayman applied H. Cartan's value distribution theory to show the non-existence of meromorphic solutions of (*) when n > (k-1)². On the other hand, known concrete examples of those solutions are not so many in the case n < (k-1)² when k > 4. In this talk, several examples will be presented and some related topics and problems will be taken up. Many of the topics overlap with those in the paper, G.G. Gundersen and W.K. Hayman, The Strength of Cartan's version of Nevanlinna Theory, Bull. London Math. Soc. 36 (2004), 433-454, but this talk will emphasize the 'extremality' of those solutions of (*), in other words of Cartan's second main theorem.

Thursday, March 17, 2005, 245 Altgeld Hall, 4:00 p.m.