Science Fair Project Encyclopedia In mathematics, specifically
in complex analysis and algebraic geometry, the Riemann-Roch
theorem is an important tool in the computation of the dimension
of the space of meromorphic functions with prescribed zeroes and
allowed poles. It relates the complex analysis of a connected
compact Riemann surface with the surface's purely topological
genus g, in a way that can be carried over into purely algebraic
settings. Initially proved as Riemann's inequality, the theorem
reached its definitive form for Riemann surfaces after work of
Riemann's student Gustav Roch in the 1850s. It was later
generalized to algebraic curves, to higher-dimensional varieties
and beyond. We start with a connected compact Riemann surface of
genus g, and a fixed point P on it. We may look at functions
having a pole only at P. There is an increasing sequence of
vector spaces: functions with no poles (i.e., constant
functions), functions allowed at most a simple pole at P,
functions allowed at most a doubl