The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is \emph{softly linear} in~$N$, i.e.~linear in~$N$ with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is \emph{purely linear} in~$N$, even in absence of FFT. The key result making this improvement possible is that the entries of the $N$th power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of~$N$. Similar algorithms are proposed for two related problems: computing the $N$th term of a C-finite sequence of polynomials, and modular exponentiation to the power $N$ for bivariate polynomials.

翻译：对于固定大小和次数的多项式矩阵的$N$次幂，可通过二进制幂运算以与乘以$N$~次线性多项式一样快的速度完成计算。当快速傅里叶变换（FFT）可用时，所得复杂度在$N$ 上为\textit{软线性}，即线性于$N$ 且含有额外的对数因子。我们证明，即使在没有FFT的情况下，也能通过一种复杂度在$N$ 上为\textit{纯线性}的算法来超越二进制幂运算。这一改进的关键在于，多项式矩阵$N$ 次幂的条目满足阶数和次数均与$N$ 无关的常系数线性微分方程。针对两个相关问题也提出了类似算法：计算C有限多项式序列的第$N$ 项，以及双变量多项式的模$N$ 次幂运算。