On an open problem related to the strict local minima of multilinear objective functions

This paper gives a combinatorial proof of a ''yes'' answer to an open question presented in [1], stated as follows: ''given a multilinear polynomial E(x): [0, 1](n) --> R, is it true that E-b(x) = E(x) - b(t)z has a strict local minimum over the discrete set {0,1}(n) for almost all b of sufficiently small norm?'' The given combinatorial proof is completed directly by providing a sufficient condition for a conjecture on the strict local minima of multilinear polynomials also postulated in [1] to hold. In addition, a simple counterexample is presented to demonstrate that the conjecture may be not true if the provided sufficient condition is not satisfied.