  Polynomial optimization problems over binary variables can be expressed as
integer programs using a linearization with extra monomials in addition to
those arising in the given polynomial. We characterize when such a
linearization yields an integral relaxation polytope, generalizing work by Del
Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and
Rodr\'iguez-Heck (European Journal of Operations Research, 2019). We also
present an algorithm that finds these extra monomials for a given polynomial to
yield an integral relaxation polytope or determines that no such set of extra
monomials exists. In the former case, our approach yields an algorithm to solve
the given polynomial optimization problem as a compact LP, and we complement
this with a purely combinatorial algorithm.
