稀疏多项式GCD算法在所有基本参数上渐近线性

中文摘要 AI 辅助

设 $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ 为整数系数的多元多项式，令 $G = \gcd(A, B)$。我们提出一种计算 $G$ 的算法，其期望位复杂度在所有基本参数上渐近线性：变量个数 $n$、项数 $T = \max\{\\|A\\|_0, \\|B\\|_0, \\|G\\|_0\}$、总次数 $D$、以及对数系数大小 $\log\Hi$ 和 $\log\Ho$，其中 $\Hi$ 界定输入系数，$\Ho$ 界定 GCD 的系数。位复杂度由简洁的界 \\[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr) \\] 刻画。据我们所知，这是首个在整数上同时实现所有这些参数线性复杂度的稀疏 GCD 算法。整数算法建立在一个新的域上 GCD 算法之上。对于域 $\K$ 上满足 $\operatorname{char}(\K) = 0$ 或 $\operatorname{char}(\K) > °G$ 的 $A, B \in \K[x_1, \dots, x_n]$，我们给出首个以期望 \\[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \\] 次域运算计算 $G = \gcd(A,B)$ 的算法，该复杂度既对输入敏感又对输出敏感。这两个算法背后的关键技术贡献是本文引入的导数辅助分离 Hensel 提升技术。通过引入辅助变量并利用导数信息，我们的方案对每个变量通过单次 $z^2$-提升提取所有部分指数，实现常数顺序深度 $O(1)$。这与经典 Hensel 提升形成鲜明对比，后者需要 $O(D)$ 次顺序提升步骤，并在稀疏设置中遭受表示稠密化问题。域算法随后通过模归约和有理重构扩展到整数情形。

英文摘要

Let $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \max\{\|A\|_0, \|B\|_0, \|G\|_0\}$, the total degree $D$, and the logarithmic coefficient sizes $\log\Hi$ and $\log\Ho$, where $\Hi$ bounds the coefficients of the inputs and $\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr). \] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \in \K[x_1, \dots, x_n]$ over a field $\K$ with $\operatorname{char}(\K) = 0$ or $\operatorname{char}(\K) > °G$, we give the first algorithm that computes $G = \gcd(A,B)$ with expected \[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.

发表机构

• Shandong University（山东大学）
• State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
• University of Chinese Academy of Sciences（中国科学院大学）

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