  We present solutions to a set of problems that arise in quantum entanglement
theory, whose common trait is the use of algebraic methods. The backbone of the
thesis consists of two general theorems, pertaining to specific convex sets of
quantum maps. They are complemented by solutions of many more specific problems
of rather technical character. These problems concerned questions such as the
higher rank numerical ranges, the lengths of separable states, the solutions of
compression equations, the completely entangled subspaces, and maximally
entangled states. The solutions were obtained partly by hand, and partly by
using the Groebner basis toolset available in computer algebra systems. The
main results, which are those of general nature, consist of a characterization
of mapping cones, which is a full analogue of the positive maps criterion for
separability, as well as of a full characterization of
positive-partial-transpose entangled states of minimum rank. The thesis also
contains an introduction to the theory of quantum entanglement and to basic
algebraic geometry, whose methods have been used
