【摘 要】
：
Vincent定理指出：给定d次实系数多项式f(x)和开区间(a1,b1).多项式f(x)在区间(a1,b1)上没有实根,当且仅当存在正常数δ,对任意区间(a,b)(∈)(a1,b1)满足|a-b|＜δ时,使得多
【机 构】
：
中科院成都计算机应用研究所,成都,610041西南民族大学计算机科学与技术学院,成都,610041
论文部分内容阅读
Vincent定理指出：给定d次实系数多项式f(x)和开区间(a1,b1).多项式f(x)在区间(a1,b1)上没有实根,当且仅当存在正常数δ,对任意区间(a,b)(∈)(a1,b1)满足|a-b|＜δ时,使得多项式(1+x)df(a+bx/1+x)的系数不变号(都是正数或都是负数).本文的主要工作是推广这一结果到一般的多变元代数系统.实系数多项式f∈R[x1,…,xn].f相对于变元xi的次数记为di.区间的笛卡尔积I=[a1,b1]×…×[an,bn](也称为Box),记φ(I)=max{bi-ai,i=1,…,n}.定义fI=(1+x1)d1…(1+xn)dnf(a1+b1x1/1+x1,…,an+bnxn/1+xn),fI称为f相对于Box I的伴随多项式.本文证明了：对给定的多项式f1,…,fm∈R[x1,…,xn],且Box ∧(∈)Rn.方程组{f1=0,…,fm=0}在Box ∧上没有零点,当且仅当存在正常数δ(与Box ∧有关),对任意Box I(∈)∧,当φ(I)＜δ时,使得伴随多项式f1I,…,fmI中至少有一个fiI的系数全是正(或负)数且fi在Box I的所有顶点上的值不为0.
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