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：我们通过对称化论证，证明了双层$\mathsf{AND}$-$\mathsf{OR}$树近似度的一个简洁且近乎紧的下界。具体而言，我们证明了$\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ \mathsf{OR}_n) = \widetilde{\Omega}(\sqrt{mn})$。与大多数将近似度表述为线性规划[BT13, She13, BDBGK18]的证明方法不同，本证明通过一系列对称化步骤将问题归约至$\mathsf{OR}$函数。我们的证明还展示了Aaronson、Kothari、Kretschmer和Thaler [AKKT19]先前提出的涉及Laurent多项式（含负指数多项式）对称化技术的强大能力。