关于多项式正、负零点个数的确定及其孤立性
On the Determination of the Number of Positive and Negative Polynomial Zeros and Their Isolation
摘要：
本文提出了一种新的双变量方法，利用该方法可以对实系数多项式$n$的正、负零点个数进行限制，其确定性明显优于笛卡尔符号法则，并且可以很好地分离出多项式的零点。该方法依赖于求解次数小于给定多项式的次数的方程。人们可以解析地确定次数高达并包括五次的多项式的正零点和负零点的确切数目，并且还可以解析地完全隔离多项式的零点，并且利用该方法的变型之一，人们可以通过求解次数不超过四次的方程来解析地接近次数高达并包括九次的多项式。对于更高次的多项式，该方法的两种变体中的任何一种都应该递归地应用。给出了三次方程根的完整分类及其隔离间隔。列举了许多例子。
英文摘要：
A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the Descartes rule of signs and also isolate quite successfully the zeros of the polynomial. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree no more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Full classification of the roots of the cubic equation, together with their isolation intervals, is presented. Numerous examples are given.
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。
