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Polynomial optimization problems over binary variables can be expressed asinteger programs using a linearization with extra monomials in addition tothose arising in the given polynomial. We characterize when such alinearization yields an integral relaxation polytope, generalizing work by DelPia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama andRodr\'iguez-Heck (European Journal of Operations Research, 2019). We alsopresent an algorithm that finds these extra monomials for a given polynomial toyield an integral relaxation polytope or determines that no such set of extramonomials exists. In the former case, our approach yields an algorithm to solvethe given polynomial optimization problem as a compact LP, and we complementthis with a purely combinatorial algorithm.

Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials