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 by Vidyasagar (ibid. vol.40, 1995), stated as follows: "Given a multilinear polynomial E(x): [0, 1]/sup n//spl rarr//spl Rscr/, is it true that E/sub b/(x)=E(x)-b/sup t/x has a strict local minimum over the discrete set {0, 1}/sup 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 Vidyasagar, to hold. In addition, a simple counter-example is presented to demonstrate that the conjecture may be not true if the provided sufficient condition is not satisfied.