We study the local decodability and (tolerant) local testability of low‐degree n‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}ⁿ. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals.

中文翻译：

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网格上多项式的本地解码和测试

我们研究了在域{0,1} ^n上评估的任意域上低阶n变量多项式的局部可解码性和（可容忍）局部可测性。我们表明，对于每个字段，都有一个容忍的本地测试，其查询复杂度仅取决于程度。相反，我们表明，在具有正特性的字段上可能存在可解码性，但在实数字段上则不存在。