Plane vs. Plane Low Degree Test

In this work, we give an optimal analysis of the plane versus plane test of Raz and Safra (STOC'97). More specifically, consider a table \(\mathcal{T}\) that assigns every plane \(P\) from \(\mathbb{F}_q^m\) a bivariate degree \(d\) polynomial. The goal is to check if these polynomials are restrictions of a global degree \(d\) polynomial \(f : \mathbb{F}_q^m \to \mathbb{F}_q\). Raz and Safra introduced the following natural test: sample two random planes \(P, P'\) intersecting in a line \(\ell\) and check if \(\mathcal{T}(P)|_{\ell} = \mathcal{T}(P')|_{\ell}\), i.e., the two table entries agree on the points on~\(\ell\).