Subresultants of (x-a)^m and (x-b)^n, Jacobi polynomials and complexity

BOSTAN, ALIN; VALDETTARO, MARCELO; KRICK, TERESA; SZANTO, AGNES

JOURNAL OF SYMBOLIC COMPUTATION

ACADEMIC PRESS LTD-ELSEVIER SCIENCE LTD

Lugar: Amsterdam; Año: 2020 vol. 101 p. 330 - 351

0747-7171

In an earlier article together with Carlos D´Andrea [BDKSV2017], we described explicit expressions for the coecients of the order-d polynomial subresultant of (x-a)^m and (x-b)^n with respect to Bernstein´s set of polynomials {(x-a)^j(x-b)^{d-j}, 0le j le d}, for 0 le d < min{m; n}. The current paper further develops the study of these structured polynomials andshows that the coecients of the subresultants of  (x-a)^m and (x-b)^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 sub resultants are scalar multiples of Jacobi polynomials up to an affine change of variables.