We prove a simple, nearly tight lower bound on the approximate degree of the two-level $\mathsf{AND}$-$\mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ \mathsf{OR}_n) = \widetilde{\Omega}(\sqrt{mn})$. We prove this lower bound via reduction to the $\mathsf{OR}$ function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [BT13, She13, BDBGK18]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson, Kothari, Kretschmer, and Thaler [AKKT19].

翻译：具体地说,我们用对称参数来证明,在两种水平的树的大约含量上,我们用对称参数来证明一个简单、近乎紧凑的下限。我们用一系列的对称步骤来证明,通过对称步骤来减少对美元对美元的作用,我们证明这一较低约束,而大多数其他证据则涉及将近似程度作为线性程序[BT13、She13、BDBGK18]。我们的证据还表明,以前由Aaronson、Kothhari、Kretschmer和Thaller[AKKT19]引进的洛朗多球(具有负功率的极人)的对称技术的力量。