多项式在有理数域上的蒙特卡洛不可约性与算术不可约性检测
Monte-Carlo Irreducibility and Imprimitivity Detection of Polynomials over $\mathbb{Q}$
AI总结：
本文提出了一种高效的蒙特卡洛算法，用于检测有理数域上多项式的算术不可约性，并通过子域提取提供构造性代数证书。
摘要：
我们研究了用于测试有理数域上多项式不可约性和检测算术不可约性的快速蒙特卡洛方法。基于Pemantle-Peres-Rivin的子集和准则，我们开发了一种概率不可约性测试，其预期运行时间（以检查的素数数量衡量）在通用输入情况下与多项式次数呈对数关系。与标准的模不可约性测试不同，该方法聚合了模因子分解的信息，而不是丢弃失败的试验。我们证明，当这种测试失败并与标准的模不可约性证书结合时，是非通用代数结构的强指示器。特别是，它经常表明伽罗瓦作用的算术不可约性。我们提出了一种显式且高效的蒙特卡洛算法，通过子域提取来检测此类不可约性，从而在不可约情况下提供构造性的代数证书。据我们所知，这是首个针对有理数域上高次多项式算术不可约性检测的实用算法。我们进一步表明，Pemantle--Peres--Rivin测试产生的子集和数据为多项式因式分解提供了预热启动，通过显著限制有理因子的可能次数，显著加速了后续的提升过程。所提出的方法在实践中比已知的确定性算法快几个数量级，并且在当前确定性技术无法达到的高次多项式中也有效。
英文摘要：
We study fast Monte-Carlo methods for testing irreducibility and detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$. Building on the subset-sum criterion of Pemantle-Peres-Rivin, we develop a probabilistic irreducibility test whose expected running time, measured in the number of primes examined, is logarithmic in the degree for generic inputs. Unlike the standard modular irreducibility test, the method aggregates information from modular factorizations rather than discarding unsuccessful trials. We show that failure of this test, when combined with a standard modular irreducibility certificate, is a strong indicator of non-generic algebraic structure. In particular, it often signals arithmetic imprimitivity of the Galois action. We present an explicit and efficient Monte-Carlo algorithm for detecting such imprimitivity via subfield extraction, yielding constructive algebraic certificates in the imprimitive case. To our knowledge, this is the first practical algorithm for detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$ in high degree. We further show that the subset-sum data produced by the Pemantle--Peres--Rivin test provides a warm start for polynomial factorization by sharply restricting the possible degrees of rational factors, significantly accelerating subsequent lifting procedures. The proposed methods are orders of magnitude faster in practice than known deterministic algorithms, and are effective in degrees far beyond the reach of current deterministic techniques.
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