In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this lower bound is actually quite a good approximation. We finish with some numerical examples indicating that these ideas may be of use in the problem of finding average Markov factors for the bounds of the uniform norm of the derivative a polynomial in terms of the norm of the polynomial itself.

中文翻译：

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关于多项式的最大值在区间端点的概率

在本文中，我们研究了在一个端点处达到最大次数为 n 次的单变量多项式的对称区间 [−a, a] 上的最大值的概率。我们给出了度数 n = 1, 2,3 的明确公式。3 级情况的公式导致真实概率的下限。数值实验表明，这个下限实际上是一个很好的近似值。我们以一些数值例子结束，表明这些想法可能在寻找平均马尔可夫因子的问题中有用，该问题是根据多项式本身的范数来求导多项式的一致范数的边界的。