Subresultants of $(x-\alpha)^m$ and $(x-\beta)^n$, Jacobi polynomials and complexity

International audienceIn an earlier article together with Carlos D'Andrea [BDKSV2017], we describedexplicit expressions for the coefficients of the order-$d$ polynomialsubresultant of $(x-\alpha)^m$ and $(x-\beta)^n $ with respect to Bernstein'sset of polynomials $\{(x-\alpha)^j(x-\beta)^{d-j}, \, 0\le j\le d\}$, for$0\le d<\min\{m, n\}$. The current paper further develops the study of thesestructured polynomials and shows that the coefficients of the subresultants of$(x-\alpha)^m$ and $(x-\beta)^n$ with respect to the monomial basis can becomputed in linear arithmetic complexity, which is faster than for arbitrarypolynomials. The result is obtained as a consequence of the amazing thoughseemingly unnoticed fact that these subresultants are scalar multiples ofJacobi polynomials up to an affine change of variables