Suppose
- ${A_j},\,{\Delta _j} \in {\mathbb C^{n \times n}},\quad\big(\,j = 0,\,1,\,2,\,\ldots,\,m\,\big)$
- ${P_\Delta }\left(\lambda\right) = \left({A_m} + {\Delta _m}\right){\lambda ^m} + \, \cdots \, + \left({A_1} + {\Delta _1}\right){\lambda ^1} + \left({A_0} + {\Delta _0}\right)$ is a matrix polynomial, and $\lambda $ is a complex variable.
Can we say that
$$\lim_{\lambda \to \infty } \dfrac{{{{\left\lvert\, \lambda\, \right\rvert}^{ mn}}}}{{\left\lvert\, {\det \big({P_\Delta }\left(\lambda\right)\big) - \det \big({A_m} + {\Delta _m}\big){\lambda ^{mn}}} \,\right\rvert}} = \infty $$