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

## TL;DR

This paper reveals that subresultants of specific polynomial pairs are scalar multiples of Jacobi polynomials, enabling faster computation of coefficients with linear complexity, advancing algebraic computation methods.

## Contribution

It demonstrates that subresultants of $(x-eta)^n$ and $(x-eta)^m$ are scalar multiples of Jacobi polynomials, allowing linear-time coefficient computation.

## Key findings

- Coefficients of subresultants can be computed in linear arithmetic complexity.
- Subresultants are scalar multiples of Jacobi polynomials after an affine change of variables.
- Faster algorithms for structured polynomial subresultants are developed.

## Abstract

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.