  In this paper we prove that the only algebraic constant mean curvature (cmc)
surfaces in R^3 of order less than four are the planes, the spheres and the
cylinders. The method used heavily depends on the efficiency of algorithms to
compute Groebner Bases and also on the memory capacity of the computer used to
do the computations. We will also prove that the problem of finding algebraic
constant mean curvature hypersurfaces in the Euclidean space completely reduces
to the problem of solving a system of polynomial equations.
