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