单因素Hensel提升及其在某些多项式直线复杂度中的应用

E. Kaltofen
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引用次数: 41

摘要

对输入次数无界的直线规划给出的多项式的某些运算,给出了建立多项式直线复杂度的三个定理。第一个定理展示了如何计算单变量的高阶偏导数。另外两个定理将输出多项式的度数作为输出程序长度的参数。首先证明了当一个直线程序计算一个多元多项式的任意次幂时,该多项式也允许多项式有界的直线计算。其次,由具有相对素数辅助因子的无除法直线规划给出的多元多项式的任何因子,在输入长度和因子的阶数上也允许长度多项式的直线计算。这个结果是基于一种新的Hensel提升过程,其中只有一个因子图像被提升回原始因子。作为一个应用,我们得到了由无除法直线程序给出的多项式的最大公约数在输入长度和其自身的度方面具有多项式的直线复杂度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Single-factor Hensel lifting and its application to the straight-line complexity of certain polynomials
Three theorems are presented that establish polynomial straight-line complexity for certain operations on polynomials given by straight-line programs of unbounded input degree. The first theorem shows how to compute a higher order partial derivative in a single variable. The other two theorems impose the degree of the output polynomial as a parameter of the length of the output program. First it is shown that if a straight-line program computes an arbitrary power of a multivariate polynomial, that polynomial also admits a polynomial bounded straight-line computation. Second, any factor of a multivariate polynomial given by a division-free straight-line program with relatively prime co-factor also admits a straight-line computation of length polynomial in the input length and the degree of the factor. This result is based on a new Hensel lifting process, one where only one factor image is lifted back to the original factor. As an application we get that the greatest common divisor of polynomials given by a division-free straight-line program has polynomial straight-line complexity in terms of the input length and its own degree.
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