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Session S34 - Symbolic and Numerical Computation with Polynomials

Wednesday, July 21, 17:30 ~ 18:00 UTC-3

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

Teresa Krick

University of Buenos Aires and CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakcd162b4645c843f770b74a8cf29e9628').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addycd162b4645c843f770b74a8cf29e9628 = 'kr&#105;ck' + '&#64;'; addycd162b4645c843f770b74a8cf29e9628 = addycd162b4645c843f770b74a8cf29e9628 + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_textcd162b4645c843f770b74a8cf29e9628 = 'kr&#105;ck' + '&#64;' + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloakcd162b4645c843f770b74a8cf29e9628').innerHTML += '<a ' + path + '\'' + prefix + ':' + addycd162b4645c843f770b74a8cf29e9628 + '\'>'+addy_textcd162b4645c843f770b74a8cf29e9628+'<\/a>';

We show that the coefficients 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. This is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are essentially scalar multiples of Jacobi polynomials.

Joint work with Alin Bostan (INRIA Saclay, France), Agnes Szanto (NCSU, USA) and Marcelo Valdettaro (UBA, Argentina).