Subresultants of (x−α)m and (x−β)n, Jacobi polynomials and complexity. (November 2020)

• Record Type: Journal Article
• Authors: Bostan, A.; Krick, T.; Szanto, A.; Valdettaro, M.
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 330
• Page End: 351
• Publication Date: 2020-11
• Subjects: 13P15 -- 15B05 -- 33C05 -- 33C45 -- 33F10 -- 68W30; Subresultants -- Algorithms -- Complexity -- Jacobi polynomials; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.10.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In an earlier article (Bostan et al., 2017 ), with Carlos D'Andrea, we described explicit expressions for the coefficients of the order- d polynomial subresultant of ( x − α ) m and ( x − β ) n with respect to Bernstein's set of polynomials { ( x − α ) j ( x − β ) d − j, 0 ≤ j ≤ d }, for 0 ≤ 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 − α ) m and ( x − β ) 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.