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.
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Alin Bostan, T Krick, A Szanto, M Valdettaro. Subresultants of $(x-\alpha)^m$ and $(x-\beta)^n$, Jacobi polynomials and complexity. Journal of Symbolic Computation, Elsevier, In press, ⟨10.1016/j.jsc.2019.10.003⟩. ⟨hal-01966640v2⟩

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