  The study of the entanglement entropy and entanglement spectrum has proven to
be very fruitful in identifying topological phases of matter. Typically, one
performs numerical studies of finite-size systems. However, there are few
rigorous results for finite-size systems. We revisit the problem of determining
the rank of the "particle entanglement spectrum" of the Laughlin states. We
reformulate the problem into a problem concerning the ideal of symmetric
polynomials that vanish under the formation of several clusters of particles.
We give an explicit generating family of this ideal, and we prove that
polynomials in this ideal have a total degree that is bounded from below. We
discuss the difficulty in proving the same bound on the degree of any of the
variables, which is necessary to determine the rank of the particle
entanglement spectrum.
