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

Alin Bostan (1) , T Krick (2) , A Szanto (3) , M Valdettaro (4)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 CONICET - Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires]
3 NC State - North Carolina State University [Raleigh]
4 FCEyN - Facultad de Ciencias Exactas y Naturales [Buenos Aires]

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.

Mots clés: Jacobi polynomials; Subresultants; algorithms; complexity

Domaines: Calcul formel [cs.SC]

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, In press. ⟨hal-01966640v1⟩