A Fast Algorithm for Computing the Truncated Resultant

TL;DR

This paper introduces a fast algorithm for efficiently computing the truncated resultant of two bivariate polynomials, significantly reducing computational complexity compared to previous methods.

Contribution

The paper presents a novel algorithm that computes the first k coefficients of the resultant in near-linear time, improving over the cubic complexity of prior algorithms.

Findings

Achieves O~(kd) complexity for truncated resultant computation

Reduces computational effort compared to previous O~(d^3) algorithms

Applicable to polynomials over any field K with degree at most d

Abstract

Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R $\in$ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O~(kd) arithmetic operations in K, where the O~ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d^3) operations before computing its first k coefficients.