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New algorithms are presented for computing annihilating polynomials ofToeplitz, Hankel, and more generally Toeplitz+ Hankel-like matrices over afield. Our approach follows works on Coppersmith's block Wiedemann method withstructured projections, which have been recently successfully applied forcomputing the bivariate resultant. A first baby-step/giant step approach --directly derived using known techniques on structured matrices -- gives arandomized Monte Carlo algorithm for the minimal polynomial of an $n\times n$Toeplitz or Hankel-like matrix of displacement rank $\alpha$ using $\tildeO(n^{\omega - c(\omega)} \alpha^{c(\omega)})$ arithmetic operations, where$\omega$ is the exponent of matrix multiplication and $c(2.373)\approx 0.523$for the best known value of $\omega$. For generic Toeplitz+Hankel-like matricesa second algorithm computes the characteristic polynomial in $\tildeO(n^{2-1/\omega})$ operations when the displacement rank is consideredconstant. Previous algorithms required $O(n^2)$ operations while the exponentspresented here are respectively less than $1.86$ and $1.58$ with the best knownestimate for $\omega$.

Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices