We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tilde{\Omega}(\sqrt{\log n})$. 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：我们明确构造了首个关于 $\mathbb{F}_2^n$ 上次数 $d \ge 2$ 多项式源的非平凡提取器。该提取器所需的最小熵为 $k\geq n - \tilde{\Omega}(\sqrt{\log n})$。此前，即使对于最小熵 $k\geq n-1$ 的情况，也未有任何已知构造。本构造的一个关键要素是输入归约引理，它允许我们假设任何最小熵为 $k$ 的多项式源均可由 $O(k)$ 个均匀随机比特生成。我们还提供了强有力的形式化证据，表明多项式源的提取问题异常困难：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，我们证明合取提取器甚至无法分散最小熵 $k\geq n-O(n/\log\log n)$ 的二次多项式源。事实上，这一不可行性结果甚至适用于我们引入的一个更专门的源族——多项式非遗忘比特固定（NOBF）源。多项式 NOBF 源是一个自然的新型代数源族，位于多项式源与簇源的交集处，因此我们的不可行性结果同时适用于这两种经典场景。这尤其令人惊讶，因为我们确实拥有略高于此屏障的簇提取器——这表明合取提取器并非无种子提取领域的万能方案。