  The $k$-QSAT problem is a quantum analog of the famous $k$-SAT constraint
satisfaction problem. We must determine the zero energy ground states of a
Hamiltonian of $N$ qubits consisting of a sum of $M$ random $k$-local rank-one
projectors. It is known that product states of zero energy exist with high
probability if and only if the underlying factor graph has a clause-covering
dimer configuration. This means that the threshold of the PRODSAT phase is a
purely geometric quantity equal to the dimer covering threshold. We revisit and
fully prove this result through a combination of complex analysis and algebraic
methods based on Buchberger's algorithm for complex polynomial equations with
random coefficients. We also discuss numerical experiments investigating the
presence of entanglement in the PRODSAT phase in the sense that product states
do not span the whole zero energy ground state space.
