## Applications of the second Nevanlinna Theorem

### Eberhard Mayerhofer

(University of Vienna)

**Abstract:**
Let K be an algebraically closed ultrametric field of arbitrary characteristic,
topologically complete. We present a recent version of the Second Nevanlinna
Theorem for meromorphic functions on K due to Boutabaa and Escassut. We present
some applications to
(1) Solvability of Equations P(f)=Q(g), where f, g meromorphic, P, Q rational
functions over K; our result also contains the "Diophantic Equation"
f^n+g^m=1, n>2, m>1, f, g entire
which has no non-constant solutions.
Interestingly, f^2+g^2=1 has no non constant entire solutions as well (this in
particular shows that the sine and cosine are not entire!). This follows
elementary.

(2) Uniqueness of meromorphic functions, that is, given a set S in K,
f, g meromorphic maps. If the preimages of S under f resp. g coincide, one may
conclude that f=g, supposed S is a set with sufficiently many points.