We study the probabilistic degree over of the OR function on n variables. For , the -error probabilistic degree of any Boolean function f : {0, 1}ⁿ → {0, 1} over is the smallest nonnegative integer d such that the following holds: there exists a distribution P of polynomials of degree at most d such that for all , we have . It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the -error probabilistic degree of the OR function is at most . Our first observation is that this can be improved to which is better for small values of . In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution P have the following special structure: where each L_i(x₁, … , x_n) is a linear form in the variables x₁, … , x_n, that is, the polynomial is a product of affine forms. We show that the -error probabilistic degree of OR when restricted to polynomials of the above form is , thus matching the above upper bound (up to poly-logarithmic factors).

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关于 OR 在实数上的概率度

我们研究了OR 函数对n 个变量的概率度。对于，任何布尔函数f的-error 概率度 ：{0, 1} ⁿ  → {0, 1} over是最小的非负整数d，使得以下成立：存在度数最多为d的多项式的分布P这样对于所有，我们有。从 Tarui ( Theoret. Comput. Sci. 1993) 和 Beigel, Reingold, and Spielman ( Proc. 6th CCC 1991) 的作品可知，OR 函数的-error 概率度至多为. 我们的第一个观察结果是，对于较小的 值，这可以改进为更好。在所有已知的对于该或函数的概率多项式（包括上述改善）的结构中，多项式P在支撑分布P具有如下特殊结构：其中每个大号_我（X ₁，...，  X _Ñ）为线性形式在变量x ₁ , … ,  x _{n 中}，即多项式是仿射形式的乘积。我们表明，当限制为上述形式的多项式时，OR的-error 概率度为，从而匹配上述上限（最多为多对数因子）。