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 arithmetic 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-recursive sequence of polynomials, and modular exponentiation to the power $N$ for bivariate polynomials.

翻译：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实现，其运算速度与两个线性次数（关于N）的多项式乘法相当。当快速傅里叶变换（FFT）可用时，所得算术复杂度为关于N的"软线性"（即线性于N且附加对数因子）。我们证明，即便在无FFT条件下，仍可通过一种复杂度为关于N的"纯线性"算法超越二进制幂运算。实现该改进的关键在于：多项式矩阵N次幂的条目满足系数阶数和次数均与N无关的线性微分方程。针对两个相关问题（计算C递归多项式序列的第N项，以及二元多项式模幂运算的N次幂）也提出了类似算法。