We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \frac{\sqrt{\log n}}{(d\log \log n)^{d/2}}$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.

翻译：我们显式构造了首个针对 $\mathbb{F}_2^n$ 上次数 $d \ge 2$ 的多项式源的非平凡提取器。该提取器所需的最小熵为 $k \geq n - \frac{\sqrt{\log n}}{(d\log \log n)^{d/2}}$。此前，即使对于最小熵 $k \geq n-1$ 的情况，也未有已知构造。我们构造的关键要素是一个输入约化引理，该引理允许我们假设任意具有最小熵 $k$ 的多项式源可由 $O(k)$ 个均匀随机比特生成。我们还提供了强有力的形式化证据，表明多项式源在提取问题上异常困难：即使我们最强大的通用提取器也无法处理最小熵低于 $k \geq n-o(n)$ 的多项式源。具体而言，我们证明了和集提取器甚至无法对最小熵 $k \geq n-O(n/\log\log n)$ 的二次多项式源进行分散。事实上，这一不可能性结果甚至适用于我们引入的一个更专门的源族——称为多项式非 oblivious 比特固定（NOBF）源。多项式 NOBF 源是一类自然的代数源新族，位于多项式源与簇源的交集中，因此我们的不可能性结果同样适用于这两个经典场景。这一点尤其令人惊讶，因为确实存在略高于这一障碍的簇提取器——这表明和集提取器并非无种子提取领域的万能解决方案。