On meromorphic solutions of <Emphasis Type="Italic">f</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">g</Emphasis><Superscript>2</Superscript> = 1

We show that meromorphic solutions f, g of f ² + g ² = 1 in C² must be constant, if f _z2 and g _z1 have the same zeros (counting multiplicities). We also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations.     

Entire solutions of eiconal type equations

The paper characterizes entire solutions to partial differential equations for polynomials p in . Received: 14 August 2006     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.Received: 14 April 2004     

Uniqueness theorems for meromorphic mappings with few hyperplanes

Let f,g be linearly nondegenerate meromorphic mappings of Cm into CPn. Let be hyperplanes in CPn in general position, such that

(a)

f−1(Hj)=g−1(Hj), for all 1?j?q,

(b)

dim(f−1(Hi)∩f−1(Hj))?m−2 for all 1?i<j?q, and

(c)

f=g on .

It is well known that if q?3n+2, then f≡g. In this paper we show that for every nonnegative integer c there exists positive integer N(c) depending only on c in an explicit way such that the above result remains valid if q?(3n+2−c) and n?N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n?0. At the same time, a big number of recent uniqueness theorems are generalized considerably.     

Irreducible induced representations of ICC-groups

This is an abbreviated version of a part of the author's doctorate thesis accepted by the Fakultät für Mathematik und Informatik der Technischen Universität München on 26 January 1990     

Difference analogues of the second main theorem for meromorphic functions in several complex variables

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard‐type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137 , 457–474 (2007); Comput. Meth. Funct. Theor. 12 , No. 1, 343–361 (2012)]. The method we used is very different from theirs.     

Geometric rigidity of transcendental meromorphic functions

The classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced. The Julia sets of geometrically tame functions are proven to be either geometrical circle (in ) or to have Hausdorff dimension strictly larger than 1. Vast classes of dynamically and geometrically tame functions are identified. The research of both authors was supported in part by the Polish KBN Grant No 2 PO3A 034 25, the Warsaw University of Technology Grant no 504G 11200043000 and by the NSF/PAN grant INT-0306004. The research of the second author was supported in part by the NSF Grant DMS 0400481.     

Gibbs state for some class of meromorphic functions

This paper is a continuation of our earlier works [1,2] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H o exp o Q, where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values ofF we define a class of summable potentials for the maps f of the punctured cylinder induced by F. We prove the existence and uniqueness of Gibbs states for these potentials.     

Sur l'équationf(x)f(y)f(−x−y)=g(x)g(y)g(−x−y)

Sans résumé

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Homology Groups of Conormal Varieties

Let G be a real Lie group acting on real analytic manifold with finitely many orbits. We prove that the characteristic cycle map is a surjective homomorphism from the K-group of G-equivariant sheaves on X to the top homology group of the conormal variety of the G-action on X. We also show that the top homology group of the G-action on X is a free -module of rank equal to the number of G-orbits. This work was completed with the support of the Ministry of Science, Education and Sport of Croatia, and Infodesign d.o.o., Varaždin.