An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$

TL;DR

This paper introduces an efficient $O(n \, \log^2 n)$ algorithm for computing Hankel determinants of rational power series, leveraging connections with Hankel continued fractions and Sturm sequences to improve computational methods.

Contribution

The paper presents a novel $O(n \log^2 n)$ algorithm for Hankel determinant computation, linking Hankel continued fractions and Sturm sequences for the first time.

Findings

The algorithm significantly reduces computational complexity.

A new theoretical connection between Sturm sequences and Hankel matrices.

Potential applications in symbolic computation and signal processing.

Abstract

Given the rational power series $h(x) = \sum_{i \geq 0} h_i x^i \in \mathbb{C}[[x]]$, the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{0 \leq i,j \leq n-1}$. We explore the relationship between the Hankel continued fraction and the generalized Sturm sequence. This connection inspires the development of a novel algorithm for computing the Hankel determinants $\{H_i(h(x))\}_{i=0}^{n-1}$ using $O(n \log^2 n)$ arithmetic operations. We also explore the connection between the generalized Sturm sequences and the signature of Hankel matrices.