## Section: New Results

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

A previous article 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 $\{{(x-\alpha )}^{j}{(x-\beta )}^{d-j},\phantom{\rule{0.166667em}{0ex}}0\le j\le d\}$, for $0\le d<min\{m,n\}$. In [3], Alin
Bostan, together with T. Krick, M. Valdettaro (U. Buenos Aires, Argentina) and
A. Szanto (U. North Carolina, Raleigh, USA) further developed the study of
these structured polynomials and showed 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.