关于Sturm定理的一点注记

A Note on Sturm Theorem

经典的S turm定理用于判定多项式在给定区间上不同的实根个数,但是并不能刻画重根的情况.在这里定义了推广的S turm序列,将S turm定理进行一定地延拓,给出区间上多项式的所有实根均是偶重根或奇重根的充要条件.作为应用,讨论了多项式正(负)半定的判定问题.

Classical Sturm theorem is used to tell the number of distinct roots of polynomials in arbitrary interval. Yet it fails to give more information such as multiplicities of roots. We define the generalized Sturm sequence, and a necessary and sufficient condition for all the roots of polynomials of any degree in given interval have even or odd multiplicity is presented. Also, we discuss the problem of the non-negative and non-positivity of polynomials as application.