Lower Bounding the AND-OR Tree via Symmetrization

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].