By W. E. Kirwan, L. Zalcman
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Additional info for Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74
Of the using extension in which of the m a n n e r ~ A; theorem of us to e x t e n d ~ This of the It is fuchsian G gives a geometrically groups step in the f o l l o w i n g give A axes to the fixed points of in the proof, theorem. G equivalent points on ~A, ~ first and perhaps of the t h e o r e m This to tes- a map defined from the h y p o t h e s i s ~A. here. of the b e h a v i o r is applied between the lattice on to the (which we prove to a r b i t r a r y of their a c c u m u l a t i o n they are located in has had great by a h o m e o m o r p h i s m .
T H E O R E M 3. If P monic in Rm where is a n o n - n e g a t i v e constant. c and is a harmonic p o l y n o m i a l in vP > 0 Rm outside a compact set, and then v is har- v = cP, These results raise two interesting open problems. Q u e s t i o n i. , such that max [xr~r Q u e s t i o n 2. log IP(x)l = o(r)? If two harmonic f u n c t i o n s v I, v 2 in Rm have a common tract, do they have the same sign everywhere? T h e o r e m 2 shows That if m : 2 the a n s w e r to both these questions is yes.
Be a positive integer Then if ~k~(1) ~ ~k (t)u(x), is a homogeneous k harmonic poly- 1 1 ~ + ~ (m-l] = harmonic p o l y n o m i a l of degree s + k in xI to Xm+ 1 9 Suppose first that ~i k degree in k is even. t 2 = x ~+I/R2 Rkp~l)(t ) is clearly Similarly a homogeneous if k Rk_(~ r k )(t) where ~ homogeneous is a polynomial polynomial It remains x = Xm+ I, : = is a polynomial of Thus 89 cv x2vm+IRk-2v ~=0 of degree k in xI to Xm+ I. k = 2p + i, RtR2P~(t 2) of degree of degree to show that p2 = [mv:l xg, 2 P~I) by Lemma 4.
Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74 by W. E. Kirwan, L. Zalcman