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In this thesis we compute the Hilbert-Kunz functions of two-dimensional ringsof type ADE by using representations of their indecomposable, maximalCohen-Macaulay modules in terms of matrix factorizations, and as first syzygymodules of homogeneous ideals. For these computations we need to compute theHilbert-series of the involved first syzygy modules as well as the fact thatthe surface rings of type ADE defined over the complex numbers are the rings ofinvariants of C[x,y] under actions defined by the finite subgroups of SL2(C).  Finally, we will show that the developed tools can be used to computeHilbert-Kunz functions of two-dimensional rings of type ADE with respect toother (monomial) ideals than the maximal. Moreover, Hilbert-Kunz functions oftwo-dimensional Fermat rings can be computed by the same method.  There is also a paper, named "The Hilbert-Kunz functions of two-dimensionalrings of type ADE", which gives a survey of main ideas and results.  Moreover, the main theorem from Monskys paper "The Hilbert-Kunz multiplicityof an irreducible trinomial" will be generalized to a non-standard gradedsituation. This result will be used to study the behavior of the Hilbert-Kunzmultiplicity in degenerations of the hypersurface rings defnied by thosespecific trinomials.  Once more I want to thank my supervisor Holger Brenner for his guidance andfriendship during the years.

Hilbert-Kunz functions of surface rings of type ADE