多项式判别矩阵的若干性质及其应用

具有文字系数的多项式f(x)，其判别矩阵是f与f′的Sylvester矩阵通过添加一行一列而得，已经知道，判别矩阵的偶数阶主子式的符号确定了f(x)的相异根(实根、复根)的数目，这里介绍如何将奇数阶与偶数阶主子式相结合用以判定该多项式的相异负根或正根的数目，并进一步判定其在区间上的实根数，本文还研究了与判别矩阵相关的一些实用性质，并应用这些性质给出了4次键合多项式不能正分解的一组简洁的充分必要条件。

SOME PROPERTIES OF THE DISCRIMINATION MATRIX OF POLYNOMIALS WITH APPLICATIONS

Given a polynomial f(x) with general symbolic coefficients, the discrimination matrix of f(x) is defined to be the Sylvester matrix of / and /'. It is well known that the even order minors of discrimination matrix determine the number of distinct (real and complex) roots of f(x). In this paper, we introduce a method which makes use of the odd order minors as well as the even order ones to determine the number of distinct negative or positive roots of f(x), and to further determine the number of roots in a given interval. Some interesting and practical properties of discrimination matrix are studied. We apply our method to the problem of determining the p-irreducibility of binding polynomials with degree 4 and obtain an explicit criterion.