  We prove that the entanglement entropy of the ground state of a locally
gapped frustration-free 2D lattice spin system satisfies an area law with
respect to a vertical bipartition of the lattice into left and right regions.
We first establish that the ground state projector of any locally gapped
frustration-free 1D spin system can be approximated to within error $\epsilon$
by a degree $O(\sqrt{n\log(\epsilon^{-1})})$ multivariate polynomial in the
interaction terms of the Hamiltonian. This generalizes the optimal bound on the
approximate degree of the boolean AND function, which corresponds to the
special case of commuting Hamiltonian terms. For 2D spin systems we then
construct an approximate ground state projector (AGSP) that employs the optimal
1D approximation in the vicinity of the boundary of the bipartition of
interest. This AGSP has sufficiently low entanglement and error to establish
the area law using a known technique.
