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 $f: \mathbb{F}_q^m \to \mathbb{F}_q$ 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 $\epsilon \geq \Omega((\frac{d}{q})^\tau))$ for some fixed $\tau > 0$, then there is a global degree-$d$ polynomial $Q: \mathbb{F}_q^m \to \mathbb{F}_q$ with agreement nearly $\epsilon$ 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 $\epsilon < \frac{1}{2}$). The previous results in this space either required $\epsilon > \frac{1}{2}$ (Polishchuk \& Spielman, STOC 1994), or $q = \Omega(d^4)$ (Arora \& Sudan, Combinatorica 2003), or needed 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 ``bootstrapping'' 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. 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.

翻译：我们证明，对于有限域上的多项式，最自然的低度数测试在“高误差”区域对于线性规模域具有“鲁棒性”。具体而言，我们考虑函数 $f: \mathbb{F}_q^m \to \mathbb{F}_q$ 与次数为 $d$ 的多项式空间的“局部”一致性，即该函数在 $\mathbb{F}_q^m$ 中随机选取的直线上与一元 $d$ 次多项式的期望一致性，并证明若该局部一致性满足 $\epsilon \geq \Omega((\frac{d}{q})^\tau))$（其中 $\tau > 0$ 为固定常数），则存在一个全局 $d$ 次多项式 $Q: \mathbb{F}_q^m \to \mathbb{F}_q$ 使得 $Q$ 与 $f$ 的一致性接近 $\epsilon$。这一结果解决了低度数测试领域长期悬而未决的问题，在“高误差”区域（即 $\epsilon < \frac{1}{2}$ 时）实现了 $O(d)$ 查询复杂度的鲁棒测试。此前该领域的结果要么要求 $\epsilon > \frac{1}{2}$（Polishchuk & Spielman, STOC 1994），要么要求 $q = \Omega(d^4)$（Arora & Sudan, Combinatorica 2003），或者需在二维“平面”（而非一维直线）上测量局部距离，导致 $\Omega(d^2)$ 查询复杂度（Raz & Safra, STOC 1997）。我们的分析遵循此前大多数分析的精神：先分析低变量情形（$m = O(1)$），再通过“自举”推广到一般多元情形。主要技术创新在于对二元情形的全新分析：以非“黑盒”方式利用多元因式分解与寻找（或测试）低度数多项式之间已知的联系。另一贡献是自举分析——该分析成功将 $m=2$ 的结果直接推广到一般 $m$ 的情形，而此前工作需从 $m=3$ 或 $m=4$ 入手——可论证地，这一自举方法显著简化了先前的工作。