Abstract: The robust oscillation for some polynomials' families is discussed. A vertex verification result for the robust oscillation of polytopic polynomial family, under some conditions, is obtained. Moreover, the number of vertex polynomial to be checked can be reduced for interval polynomial family and diamond polynomial family. A counter-example is given to show that these vertex verification results can't be directly generalized for root distribution invariance of polynomial families.

Key words: polytopic polynomial family, interval polynomial family, diamond polynomial family, robust oscillation, vertex verification

摘要： 讨论了多项式族的鲁棒振动性。针对凸多面体多项式族，指出在一定条件下，其整族多项式的鲁棒振动性可由其顶点多项式振动性检验得到，对于区间多项式族和菱形多项式族，需要检验的顶点多项式数目还可进一步减少。反例表明，这些顶点检验结果不能直接推广到多项式族的根均不在一给定直线上的情形。

关键词: 凸多面体多项式族, 区间多项式族, 菱形多项式族, 鲁棒振动性, 顶点检验