We prove that $\mathrm{deg}(f) \leq \widetilde{O}(\mathrm{rdeg}(f)^3)$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.
翻译:我们证明对任意布尔函数 $f$,有 $\mathrm{deg}(f) \leq \widetilde{O}(\mathrm{rdeg}(f)^3)$,其中 $\mathrm{deg}(f)$ 是 $f$ 的次数,$\mathrm{rdeg}(f)$ 是 $f$ 的有理度。这一结果解决了 Nisan 和 Szegedy 于 1994 年提出的、归功于 Fortnow 的三个开放问题中的第二个。