整系数多项式根的符号计算
Symbolic computation of the roots of any polynomial with integer coefficients
摘要：
具有整数系数的任何m次多项式的根可以通过操纵由2m个不同符号组成的序列并对序列中的不同符号进行计数来计算。这种方法只需要像序列替换和符号计数这样的“原始”操作。不需要使用“高级”运算（如乘法、除法、除法等）进行计算。该方法可以实现为一个尺规作图法的根的多项式，以任意精度使用一个直尺，一个指南针，和铅笔的2米不同的颜色。特别是，古老的问题“加倍的立方体”是渐近解决上述建设。这种方法，通过它可以加倍一个立方体，虽然，在无限的步骤，可能是最接近的原始问题的建设只使用一个直边和指南针在有限数量的步骤。此外，有理数域上的每一个m次多项式都可以对应一个生成整数序列的m项递归关系。一组m个这样的序列，它们一起表现出有趣的属性有关的多项式的根，可以得到，如果这些m个序列中的每一个的m个初始项被选择在一个特殊的方式使用一个矩阵与多项式。只需要计算这些整数序列中的两个就可以获得具有最大绝对值的实根。由于该方法只涉及整数，因此比使用浮点运算的传统方法更快。
英文摘要：
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive' operations like replacement of sequences and counting of symbols. No calculations using 'advanced' operations like multiplication, division, logarithms etc. are needed. The method can be implemented as a geometric construction of roots of polynomials to arbitrary accuracy using only a straight edge, a compass, and pencils of 2m different colors. In particular, the ancient problem of the "doubling of cube" is soluble asymptotically by the above-mentioned construction. This method, by which a cube can be doubled, albeit, in infinite steps, is probably the closest to the original problem of construction using only a straight edge and compass in a finite number of steps. Moreover, to every polynomial of degree m over the field of rationals, can be associated an m-term recurrence relation for generating integer sequences. A set of m such sequences, which together exhibit interesting properties related to the roots of the polynomial, can be obtained if the m initial terms of each of these m sequences is chosen in a special way using a matrix associated with the polynomial. Only two of these integer sequences need to be computed to obtain the real root having the largest absolute value. Since this method involves only integers, it is faster than the conventional methods using floating-point arithmetic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
