An Improved Line-Point Low-Degree Test

We prove that the most natural low-degree test for polynomials over finite fields is “robust” in the high-error regime for linear-sized fields. Specifically we consider the “local” agreement of a function$f:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$from the space of degree-d polynomials, i.e., the expected agreement of the function from univariate degree-d polynomials over a randomly chosen line in$\mathbb{F}_{q}^{m}$, and prove that if this local agreement is$\varepsilon\geq\Omega((d/q)^{\tau}))$for some fixed$\tau > 0$, then there is a global degree-d polynomial$Q:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$with agreement nearly$\varepsilon$with$f$. This settles a long-standing open question in the area of low-degree testing, yielding an$O(d)$-query robust test in the “high-error” regime (i.e., when$\varepsilon < 1/2)$. The previous results in this space either required$\varepsilon > 1/2$(Polishchuk & Spielman, STOC 1994), or$q=\Omega(d^{4})$(Arora & Sudan, Combinatorica 2003), orneeded to measure local distance on 2-dimensional “planes” rather than one-dimensional lines leading to$\Omega(d^{2})$-query complexity (Raz & Safra, STOC 1997). Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case$(m=O(1))$and then “boot-strapping” to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non “black-box” manner. This connection was used roughly in a black-box manner in the work of Arora & Sudan — and we show that opening up this black box and making some delicate choices in the analysis leads to our essentially optimal analysis. A second contribution is a bootstrapping analysis which manages to lift analyses for$m=2$directly to analyses for general$m$, where previous works needed to work with$m=3$or$m=4$— arguably this bootstrapping is significantly simpler than those in prior works.