In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of the authors [ABPSS24] (STOC 2024) who considered the case of linear polynomials and gave analogous results. Low-degree polynomials over the Boolean cube $\{0,1\}^{n}$ arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABPSS24] from linear to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-$d$ polynomials. To show that the class of degree-$d$ polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [ABPSS24] for higher degree polynomials involves understanding random restrictions of non-zero degree-$d$ polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices.

翻译：在本研究中，我们证明了将 $\{0,1\}^{n}$ 映射到任意阿贝尔群 $G$ 的多元 $d$ 次多项式类，可在查询次数为 $\widetilde{O}_{d}((\log n)^{d})$ 的条件下进行局部校正，其可容忍的错误比例接近底层码最小距离的一半。特别地，这一结论对于实数域或有理数域上的多项式同样成立——这两种特殊情况此前并未被知晓。此外，我们还证明了它们可在错误比例接近码的最小距离时进行局部列表校正。这些结果建立并扩展了作者先前的工作 [ABPSS24]（STOC 2024），该工作考虑了线性多项式的情形并给出了类似结论。布尔立方 $\{0,1\}^{n}$ 上的低次多项式自然地出现在布尔电路复杂性与学习理论中，我们的工作进一步推进了对其编码理论性质的研究。将 [ABPSS24] 的结果从线性情形推广到更高次多项式面临若干新的挑战，处理这些挑战使我们更深入地理解了布尔立方上低次多项式的性质。对于局部校正，我们构造了布尔立方中的一个点集，该点集位于两个指数级接近的平行超平面之间，并且是 $d$ 次多项式的插值集。为了证明 $d$ 次多项式类可列表解码至最小距离，我们综合运用了关于低次多项式反集中性的结果、Sunflower 引理，以及用于计算多项式公共零点个数的 Footprint 界。分析 [ABPSS24] 中针对高次多项式的局部列表校正器，需要理解非零 $d$ 次多项式在汉明切片上的随机限制。特别地，我们证明了一个用于降低布尔立方维度的简单随机限制过程，是汉明切片的一个足够良好的采样器。