整数系数多项式最大公因式的一种新稀疏算法
A New Sparse Algorithm for Polynomial GCD over Integers
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中文总结 AI 辅助
本文提出一种新的稀疏算法，通过新变量替换将多元整数多项式GCD归约为单个单变量GCD，位复杂度为多项式级，在Maple中实现，对高次多变量但GCD项数少的多项式高效。
中文摘要 AI 辅助
我们描述了一种新的用于整数系数多项式的最大公因式（GCD）算法。新算法的位复杂度关于输入和输出大小以及单个次数是多项式的。该算法遵循标准方法，将多元多项式GCD归约为单变量多项式GCD。我们的算法将多元多项式GCD归约为单个单变量多项式GCD。我们算法的主要思想是一种新的变量替换，它将多元多项式化为分离多项式，即主变量中的系数均为单项式。我们分析了显式的位复杂度，并在Maple中实现了我们的算法。结果表明，我们的算法对于高次数、多变量但GCD项数较少的多项式是高效的。
英文摘要
We describe a new greatest common divisor (GCD) algorithm for polynomials with integer coefficients. The bit complexity of the new algorithm is polynomial in the input and output sizes and the individual degree bounds.Our algorithm follows the standard approach by reducing multivariate polynomial GCD to univariate polynomial GCD. Our algorithm reduces a multivariate polynomial GCD to a single univariate polynomial GCD. The main idea of our algorithm is a new variable substitution which reduces a multivariate polynomial to a separated one, that is, the coefficients in a main variable are all monomials. The explicit bit complexity is analyzed and we have implemented our algorithm in Maple. It is shown that our algorithm is efficient for polynomials with high degree, large number of variables, but small number of terms in GCD.
发表机构
- Shandong University（山东大学）
- Simon Fraser University（西蒙菲莎大学）
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