Optimal bound for the polynomial Littlewood-Offord problem

We present an exposition of an argument, discovered by GPT-6 Pro, that gives an optimal bound for the polynomial Littlewood-Offord problem. Namely, let $F$ be a degree-$d$ multilinear polynomial that contains $r$ degree-$d$ monomials involving disjoint sets of variables. Then, for i.i.d. Rademacher random variables ${\xi }_1,\dots ,{\xi }_n$, we have $\mathbb{P}[F({\xi }_1,\dots ,{\xi }_n)=0]=O_d(r^{-1\mathrm{/}2})$. This improves upon the previous bound of $(\log r{)}^{O_d(1)}{r}^{-1\mathrm{/}2}$ due to Meka, O. Nguyen, and Vu, and resolves a conjecture attributed to H. Nguyen and Vu. The key part of the proof is an estimate for the total influence of bounded-degree rational functions, which resolves a recent conjecture of Kothari, Kovacs-Deak, Wang, and Yang.