# Lower Bounding the AND-OR Tree via Symmetrization

## TL;DR

This paper establishes a nearly tight lower bound on the approximate degree of the AND-OR tree using symmetrization techniques, advancing understanding of polynomial approximation complexity.

## Contribution

It introduces a novel symmetrization approach involving Laurent polynomials to prove lower bounds on approximate degree, differing from previous linear programming methods.

## Key findings

- Proves ( ext{AND}_m \u22ef ext{OR}_n) = \u223c \u221a{mn}
- Demonstrates the effectiveness of Laurent polynomial symmetrization techniques
- Provides a simpler, more direct proof compared to prior linear programming approaches

## Abstract

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].