Subresultants of (x−α)m and (x−β)n, Jacobi polynomials and complexity

Abstract

In an earlier article (Bostan et al., 2017), with Carlos D’Andrea, we described explicit expressions for the coefficients of the order-d polynomial subresultant of (x − α) m and (x − β)n with respect to Bernstein’s set of polynomials {(x − α)j (x − β)d− j , 0 ≤ j ≤ d}, for 0 ≤ 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 − α) m and (x − β)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.