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.
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机译：本文给出了Vidyasagar（同上，第40卷，1995年）提出的一个开放问题的“是”答案的组合证明，其陈述如下：“给出多元线性多项式E（x）：[0，1] / sup n // spl rarr // spl Rscr /，确实是E / sub b /（x）= E（x）-b / sup t / x在离散集{0，1} /上具有严格的局部最小值几乎所有的b都足够小？给定的组合证明是通过提供充足的条件来猜想的，该条件足以猜想多元线性多项式的严格局部极小值（也假定在Vidyasagar中）可以成立。此外，提出了一个简单的反例，以证明如果不满足提供的充分条件，则猜想可能不成立。
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