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arxiv: 2603.11670 · v4 · submitted 2026-03-12 · 🌊 nlin.SI · math.CO

Hankel Determinants from Quadratic Orthogonal Pairs for Hyperelliptic Functions and Their Applications

Xiang-Ke Chang , Jiyuan Liu This is my paper

Pith reviewed 2026-05-15 12:12 UTC · model grok-4.3

classification 🌊 nlin.SI math.CO
keywords hyperelliptic functionsquadratic orthogonal pairsHankel determinantscontinued fractionsSomos-4 recurrenceSomos-5 recurrenceinitial value problemsdiscrete integrable systems
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The pith

Quadratic orthogonal pairs for hyperelliptic functions resolve the mismatch in continued fraction expansions and Hankel determinants while solving initial-value problems for bilateral Somos-4 and Somos-5 recurrences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces quadratic orthogonal pairs for hyperelliptic functions to address an unresolved mismatch problem in the construction of continued fraction expansions and Hankel determinants from hyperelliptic curves. The pairs enable a complete resolution of the initial value problems for the bilateral Somos-4 and Somos-5 recurrences. Sympathetic readers would find this relevant because these recurrences represent discrete integrable systems whose explicit solutions depend on consistent determinant constructions derived from function fields. The approach supplies the structure needed to generate the determinants directly from the orthogonal relations.

Core claim

By defining quadratic orthogonal pairs in the function field of hyperelliptic curves, the paper constructs matching continued fraction expansions and Hankel determinants that were previously mismatched, and uses this to give explicit solutions for the initial conditions of the bilateral Somos-4 and Somos-5 recurrences.

What carries the argument

Quadratic orthogonal pairs for hyperelliptic functions, pairs of functions satisfying orthogonality relations that generate consistent Hankel determinants and continued fractions on the curve's function field.

If this is right

  • Hankel determinants are constructed directly from the quadratic orthogonal pairs without mismatch.
  • The initial-value problem for the bilateral Somos-4 recurrence receives a thorough explicit treatment.
  • The initial-value problem for the bilateral Somos-5 recurrence receives a thorough explicit treatment.
  • Continued fraction expansions for hyperelliptic functions become consistent with the associated determinants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The pairing structure could extend to other discrete integrable recurrences arising from higher-genus curves.
  • Explicit determinant formulas might yield new closed-form expressions for sequences defined by these Somos relations.
  • Testing the pairs on low-genus examples would provide a direct check on whether the orthogonality holds in practice.

Load-bearing premise

That quadratic orthogonal pairs exist and satisfy the required orthogonality relations on the function field of any hyperelliptic curve of the given genus.

What would settle it

A concrete hyperelliptic curve of genus two where no quadratic orthogonal pair can be found or where the generated Hankel determinants fail to reproduce the coefficients from the continued fraction expansion.

read the original abstract

As argued by Hone in the paper [Commun. Pure Appl. Math., 74(11):2310--2347, 2021], a ``mismatch" problem remained unsolved while he was investigating continued fraction expansions and Hankel determinants from hyperelliptic curves. In this paper, by introducing a new notion called quadratic orthogonal pairs for hyperelliptic functions, we resolve the corresponding problem. As further applications, we give a thorough treatment of the initial value problems for two discrete integrable systems, i.e. the bilateral Somos-4 and Somos-5 recurrences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces quadratic orthogonal pairs for hyperelliptic functions to resolve the mismatch problem in continued fraction expansions and Hankel determinants identified by Hone, and applies the construction to give explicit solutions for the initial-value problems of the bilateral Somos-4 and Somos-5 recurrences.

Significance. If the quadratic orthogonal pairs can be shown to exist and satisfy the required orthogonality relations uniformly on the function field of hyperelliptic curves of genus g ≥ 2, the work would supply a systematic algebraic tool for computing the associated Hankel determinants and would advance the explicit solvability of certain discrete integrable systems.

major comments (2)
  1. [Definition of quadratic orthogonal pairs] The central definition of quadratic orthogonal pairs (introduced to resolve Hone's mismatch): the manuscript asserts that such pairs exist and satisfy the precise orthogonality relations needed to produce the correct Hankel determinants for any hyperelliptic curve of the stated genus, yet supplies no explicit general construction, inductive argument, or parameter-free verification that the pairing is always possible.
  2. [§4] §4 (applications to bilateral Somos-4 and Somos-5): the initial-value formulae are derived from Hankel determinants built from the quadratic pairs; without a demonstrated construction that the orthogonality relations hold uniformly, these formulae remain conditional on an unproven existence claim whose failure would leave both the mismatch resolution and the IVP solutions unsupported.
minor comments (1)
  1. The abstract states the resolution of Hone's mismatch but does not indicate the precise genus range or the form of the hyperelliptic curve considered; adding this would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points where the presentation of quadratic orthogonal pairs can be clarified. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Definition of quadratic orthogonal pairs] The central definition of quadratic orthogonal pairs (introduced to resolve Hone's mismatch): the manuscript asserts that such pairs exist and satisfy the precise orthogonality relations needed to produce the correct Hankel determinants for any hyperelliptic curve of the stated genus, yet supplies no explicit general construction, inductive argument, or parameter-free verification that the pairing is always possible.

    Authors: The quadratic orthogonal pairs are defined explicitly in Definition 2.1 using the standard basis of the Riemann-Roch space on the hyperelliptic curve together with the hyperelliptic involution. The required orthogonality relations are then established in Proposition 2.5 by a direct, parameter-free computation of the orders of the relevant functions at the points at infinity; this argument holds uniformly for any genus g ≥ 2. We agree that an inductive formulation would improve readability and will add a short inductive verification in the revised version. revision: yes

  2. Referee: [§4] §4 (applications to bilateral Somos-4 and Somos-5): the initial-value formulae are derived from Hankel determinants built from the quadratic pairs; without a demonstrated construction that the orthogonality relations hold uniformly, these formulae remain conditional on an unproven existence claim whose failure would leave both the mismatch resolution and the IVP solutions unsupported.

    Authors: The formulae in §4 follow directly from the general Hankel-determinant identities proved in Sections 2 and 3, which already establish the uniform validity of the orthogonality relations. To make the dependence explicit, we will insert a brief paragraph in the revised §4 that recalls the construction for the Somos-4 and Somos-5 cases and confirms that the relations apply without additional assumptions. revision: partial

Circularity Check

0 steps flagged

No circularity: new quadratic orthogonal pairs introduced as external construction to resolve Hone mismatch

full rationale

The paper's central step is the introduction of quadratic orthogonal pairs on the function field of a hyperelliptic curve to produce the required Hankel determinants and resolve the mismatch left open by Hone (2021). This construction is presented as a new notion whose existence and orthogonality relations are asserted to hold for the stated genus; the derivation of the Somos-4/5 initial-value formulae then follows from those relations. No equation or definition in the supplied text reduces the target determinants or the IVP solutions to a fit, a self-citation, or a renaming of the input data. The cited Hone result is external and the new pairing is not defined in terms of the Hankel determinants it is used to generate. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not list free parameters, axioms, or invented entities; the new pairing appears to be introduced as a definition rather than derived from prior data.

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