Beating Binary Powering for Polynomial Matrices.

The Nth 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 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 purely linear in N, even in absence of FFT. The key result making this improvement possible is that the entries of the Nth 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 Nth term of a C-finite sequence of polynomials, and modular exponentiation to the power N for bivariate polynomials.