Subresultants of $(x-α)^m$ and $(x-β)^n$, Jacobi polynomials and complexity

Abstract: In an earlier article together with Carlos D'Andrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-\alpha)^m$ and $(x-\beta)^n $ with respect to Bernstein's set 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 these structured polynomials and shows that the coefficients of the subresultants of $(x-\alpha)^m$ and $(x-\beta)^n$ with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.