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

TL;DR

This work addresses the efficient computation of Hankel determinants $H_n(h(x))$ for rational power series by linking Hankel determinants to an $H$-continued fraction through a generalized Sturm sequence. The authors equip this connection with a practical, $O(n \log^2 n)$ algorithm (CompHD) that constructs an appropriate pair $(f_0,f_1)$ from $h(x)=N(x)/D(x)$, computes the Hankel continued fraction modulo $x^{2n-1}$ via half-GCD, and assembles the determinant sequence without requiring non-singularity. A key theoretical contribution is a closed-form, recursive description of $H_r$ in terms of Sturm data, enabling efficient determinant computation and yielding a robust method for rational series. Additionally, the paper derives a signature formula for Hankel matrices using the generalized Sturm sequence, connecting determinant signs to root-counting principles from Sturm theory. Overall, the results provide a fast, structure-exploiting approach to Hankel determinants and deepen the link between continued fractions, polynomial remainder sequences, and real-root counting in the Hankel context.

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.

Theorems & Definitions (24)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• Remark 3.2
• Lemma 3.3
• proof