  We explore the feasibility of interference alignment in signal vector space
-- based only on beamforming -- for K-user MIMO interference channels. Our main
contribution is to relate the feasibility issue to the problem of determining
the solvability of a multivariate polynomial system, considered extensively in
algebraic geometry. It is well known, e.g. from Bezout's theorem, that generic
polynomial systems are solvable if and only if the number of equations does not
exceed the number of variables. Following this intuition, we classify signal
space interference alignment problems as either proper or improper based on the
number of equations and variables. Rigorous connections between feasible and
proper systems are made through Bernshtein's theorem for the case where each
transmitter uses only one beamforming vector. The multi-beam case introduces
dependencies among the coefficients of a polynomial system so that the system
is no longer generic in the sense required by both theorems. In this case, we
show that the connection between feasible and proper systems can be further
strengthened (since the equivalency between feasible and proper systems does
not always hold) by including standard information theoretic outer bounds in
the feasibility analysis.
