Subresultants of (x−α)m and (x−β)n, Jacobi polynomials and complexity. (November 2020) - Bostan, A.; Krick, T.; Szanto, A.; Valdettaro, M.
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.