Subresultants of $(x-\alpha {)}^m$ and $(x-\beta {)}^n$, 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.

34 pages, accepted for publication in Journal of Symbolic Computation