碰撞和单元唯一性的对偶多项式
Dual Polynomials for Collision and Element Distinctness
摘要：
布尔函数$f: \{-1, 1\}^n \to \{-1, 1\}$的近似次数是在$\ell_\infty$范数中将$f$近似到误差$1/3$内的实多项式的最小次数。在一个有影响力的结果中，Aaronson和Shi（J. ACM 2004）分别证明了碰撞函数和元素区分函数的近似度的紧$\tildeΩ(n^{1/3})$和$\tildeΩ(n^{2/3})$下界。他们的证明是非建设性的，使用了复杂的对称化论点和近似理论的工具。近年来，近似度研究中的几个公开问题通过构造对偶多项式得到了解决。这些是一个适当的线性规划的显式对偶解，该线性规划捕获任何函数的近似度。我们通过构造碰撞函数和元素区别函数的显式对偶多项式来改进Aaronson和Shi的结果。
英文摘要：
The approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\ell_\infty$ norm. In an influential result, Aaronson and Shi (J. ACM 2004) proved tight $\tildeΩ(n^{1/3})$ and $\tildeΩ(n^{2/3})$ lower bounds on the approximate degree of the Collision and Element Distinctness functions, respectively. Their proof was non-constructive, using a sophisticated symmetrization argument and tools from approximation theory. More recently, several open problems in the study of approximate degree have been resolved via the construction of dual polynomials. These are explicit dual solutions to an appropriate linear program that captures the approximate degree of any function. We reprove Aaronson and Shi's results by constructing explicit dual polynomials for the Collision and Element Distinctness functions.
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