  Determining the feasibility conditions for vector space interference
alignment in the K-user MIMO interference channel with constant channel
coefficients has attracted much recent attention yet remains unsolved. The main
result of this paper is restricted to the symmetric square case where all
transmitters and receivers have N antennas, and each user desires d transmit
dimensions. We prove that alignment is possible if and only if the number of
antennas satisfies N>= d(K+1)/2. We also show a necessary condition for
feasibility of alignment with arbitrary system parameters. An algebraic
geometry approach is central to the results.
