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