Conserved quantities of discretizations by polarization

Noah Gie{\ss}ing; Yuri B. Suris

arxiv: 2606.29263 · v1 · pith:DOXQMIJAnew · submitted 2026-06-28 · 🌊 nlin.SI · cs.NA· math.NA

Conserved quantities of discretizations by polarization

Noah Gie{\ss}ing , Yuri B. Suris This is my paper

Pith reviewed 2026-06-30 02:08 UTC · model grok-4.3

classification 🌊 nlin.SI cs.NAmath.NA

keywords polarization discretizationconserved quantitieshigher-order ODEsalgebraic approachintegrals of motionKahan methodHamiltonian systemsvolume form

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The pith

The algebraic approach to conserved quantities in polarization discretizations extends to arbitrary order, producing new integrals for orders three and higher.

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

This paper extends an algebraic method for finding integrals of motion in discretizations obtained by polarizing vector fields. The method, already known to work for first- and second-order ODEs with polynomial right-hand sides, is shown to apply equally to ODEs of any higher order. As a result, the discretizations possess conserved quantities that are new when the order is at least three. A reader would care because these discretizations are known to preserve important geometric features such as integrals and volume forms, and confirming this for all orders makes them more reliable for simulating complex dynamical systems.

Core claim

The discretization by polarization of ODEs of arbitrary order admits an integral of motion derived by the same algebraic procedure used in low orders. For Hamiltonian systems this yields a conserved quantity together with an invariant volume form. These integrals are new for all orders greater than or equal to three.

What carries the argument

The algebraic derivation of integrals from the polarized discretization, which works by constructing suitable expressions that remain constant along the discrete flow.

If this is right

Discretizations of higher-order Hamiltonian ODEs possess an integral of motion.
They also possess an invariant volume form.
The integrals for orders three and above are previously unknown.
The conservation holds for all polynomial vector fields of the given order.

Where Pith is reading between the lines

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

These new integrals may enable more accurate long-term numerical simulations of high-order systems without artificial dissipation.
The algebraic technique could be tested on specific examples like third-order nonlinear oscillators to verify the integrals explicitly.
Similar extensions might apply to other structure-preserving discretizations beyond polarization.

Load-bearing premise

The algebraic identities that produce the conserved quantities in low orders continue to hold unchanged when the order of the ODE increases.

What would settle it

Explicit computation of the proposed integral for a cubic vector field in a third-order ODE and checking whether its value is exactly preserved after one step of the polarization discretization.

read the original abstract

Recently, a family of unconventional integrators for higher order ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for first order ODEs with quadratic vector fields. All these integrators possess remarkable conservation properties. In particular, for the first and the second order Hamiltonian ODEs, the discretization by polarization possesses an integral of motion and an invariant volume form. In this note, we extend our previously proposed algebraic approach to derivation of these integrals to discretizations of ODEs of an arbitrary order. For all orders $\ge 3$, these integrals are new.

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.

Desk Editor's Note private letter to a colleague

This note generalizes the authors' algebraic method for conserved quantities in polarization discretizations from low orders to arbitrary order and flags the higher-order integrals as new.

read the letter

The main takeaway is that the authors take their prior algebraic construction for deriving integrals of motion and invariant volume forms in polarization-based integrators and claim it works for ODEs of any order, with genuinely new integrals appearing at order 3 and above.

What the paper does is straightforward: it supplies a uniform algebraic route to these conserved quantities instead of case-by-case checks. That is useful inside the geometric integration community because it turns a collection of low-order examples into a general statement. The abstract is clear that the integrals themselves are new for the higher orders, which separates this note from the authors' earlier papers on orders 1 and 2.

The soft spot is that the abstract gives no explicit form of the generalized integral, no sample calculation for order 3 or 4, and no argument that the polynomial identities used at low order continue to close without extra hypotheses on the vector field. The stress-test concern is therefore on point: if the polarization map or the degree counting changes character at higher order, the conservation property might need additional conditions that are not addressed here. Without the derivation steps, it is impossible to judge whether the extension is mechanical or requires new work.

This is a short, technical note aimed at specialists already familiar with Kahan-type and polarization discretizations. A reader working on structure-preserving methods for polynomial systems will find the claim worth checking, but the paper does not stand on its own for a broader audience.

It is worth sending to peer review. The core claim is narrow enough that referees can verify it quickly once the full algebraic steps are supplied, and the result, if correct, removes the need for order-by-order proofs in this family of integrators.

Referee Report

2 major / 0 minor

Summary. The manuscript extends an algebraic approach, previously applied to polarized discretizations of first- and second-order Hamiltonian ODEs with polynomial vector fields, to arbitrary order. It asserts that the resulting integrals of motion and invariant volume forms are new for all orders ≥3.

Significance. If the claimed extension holds and produces genuinely conserved quantities without new assumptions, the work would supply a uniform algebraic route to invariants for a family of unconventional integrators beyond the Kahan case, strengthening the link between polarization-based discretizations and integrability in higher-order systems.

major comments (2)

[Abstract / main text] The manuscript states that the algebraic approach is extended to arbitrary order and that the integrals are new for orders ≥3, yet supplies neither the generalized construction, the explicit form of any integral for order ≥3, nor a verification that the low-order polynomial identities continue to close. This leaves the central claim unverified against the paper's own data or steps.
[Abstract] The skeptic's concern is realized: the abstract refers to extending 'our previously proposed algebraic approach' but does not demonstrate that the polarization map or underlying identities extend verbatim to higher order while preserving conservation, nor does it rule out the need for additional degree or structural hypotheses.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised identify areas where the presentation of the generalized construction can be strengthened for clarity and verifiability. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / main text] The manuscript states that the algebraic approach is extended to arbitrary order and that the integrals are new for orders ≥3, yet supplies neither the generalized construction, the explicit form of any integral for order ≥3, nor a verification that the low-order polynomial identities continue to close. This leaves the central claim unverified against the paper's own data or steps.

Authors: We agree that the current version does not supply the explicit generalized construction or examples for orders ≥3. The manuscript is a short note whose primary contribution is the statement of the extension, but to make the claim verifiable we will add, in the revised version, the full algebraic construction for arbitrary order, the explicit form of the new integrals at order 3, and a direct check that the relevant polynomial identities close under the polarization map. revision: yes
Referee: [Abstract] The skeptic's concern is realized: the abstract refers to extending 'our previously proposed algebraic approach' but does not demonstrate that the polarization map or underlying identities extend verbatim to higher order while preserving conservation, nor does it rule out the need for additional degree or structural hypotheses.

Authors: The abstract is necessarily concise. The polarization map is applied verbatim to the higher-order case exactly as in the first- and second-order settings, and the algebraic identities are shown to close without extra hypotheses on degree or structure. To remove any ambiguity, the revision will include an explicit paragraph demonstrating that the map and the conservation identities extend directly, with no additional assumptions required. revision: yes

Circularity Check

0 steps flagged

No circularity: extension of algebraic method presented as self-contained derivation for higher orders

full rationale

The paper states it extends its own prior algebraic approach (for orders 1-2) to arbitrary order and asserts the resulting integrals are new for orders >=3. No quoted step reduces a claimed conserved quantity or uniqueness result to a fitted parameter, a self-citation chain, or an input by construction; the generalized construction itself constitutes the content of the note. The self-reference is limited to motivating the extension and does not bear the load of proving conservation for the new cases.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities. All arrays are therefore empty.

pith-pipeline@v0.9.1-grok · 5634 in / 1047 out tokens · 26565 ms · 2026-06-30T02:08:53.653156+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references

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Celledoni, R

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R.I. McLachlan, D.I. McLaren, G.R.W. Quispel.Birational maps from polarization and the preservation of measure and integrals, J. Phys. A56(2023), 365202, 15 pp

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Petrera, A

M. Petrera, A. Pfadler, Yu.B. Suris.On integrability of Hirota-Kimura-type discretizations: experimental study of the discrete Clebsch system, Experiment. Math.18(2009), no.2, 223– 247

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M. Petrera, A. Pfadler, Yu.B. Suris.On integrability of Hirota-Kimura type discretizations, Regular Chaotic Dyn.16(2011), No. 3-4, p. 245–289

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Petrera, J

M. Petrera, J. Smirin, Yu.B. Suris.Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. Royal Soc. A,475(2019), 20180761, 13 pp

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Petrera, Yu.B

M. Petrera, Yu.B. Suris.New results on integrability of the Kahan-Hirota-Kimura dis- cretizations. - In:Nonlinear Systems and Their Remarkable Mathematical Structures, Ed. N. Euler, CRC Press, Boca Raton FL, 2018, p. 94–120

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Suris.A new approach to integrals of discretizations by polarization

Yu.B. Suris.A new approach to integrals of discretizations by polarization. Open Commun. in Nonlin. Math. Phys., 2024, Special Issue in Memory of Decio Levi, 11571, 8 pp. 11

2024

[1] [1]

Celledoni, R.I

E. Celledoni, R.I. McLachlan, B. Owren, G.R.W. Quispel.Geometric properties of Kahan’s method, J. Phys. A46(2013), 025201, 12 pp

2013

[2] [2]

Celledoni, R.I

E. Celledoni, R.I. McLachlan, D.I. McLaren, B. Owren, G.R.W. Quispel.Integrability properties of Kahan’s method, J. Phys. A47(2014), 365202, 20 pp

2014

[3] [3]

Celledoni, R

E. Celledoni, R. McLachlan, D. McLaren, B. Owren, G.R.W. Quispel.Discretization of polynomial vector fields by polarization, Proc. Royal Soc. A.471(2015), no.2184, 20150390, 10 pp

2015

[4] [4]

Hone, G.R.W

A.N.W. Hone, G.R.W. Quispel.Analogues of Kahan’s method for higher order equations of higher degree.– In:Asymptotic, algebraic and geometric aspects of integrable systems, Springer Proc. Math. Stat.,338, Springer, Cham, 2020, 175–189. 10

2020

[5] [5]

Kahan.Unconventional numerical methods for trajectory calculations, Unpublished lec- ture notes, 1993

W. Kahan.Unconventional numerical methods for trajectory calculations, Unpublished lec- ture notes, 1993

1993

[6] [6]

McLachlan, D.I

R.I. McLachlan, D.I. McLaren, G.R.W. Quispel.Birational maps from polarization and the preservation of measure and integrals, J. Phys. A56(2023), 365202, 15 pp

2023

[7] [7]

Petrera, A

M. Petrera, A. Pfadler, Yu.B. Suris.On integrability of Hirota-Kimura-type discretizations: experimental study of the discrete Clebsch system, Experiment. Math.18(2009), no.2, 223– 247

2009

[8] [8]

Petrera, A

M. Petrera, A. Pfadler, Yu.B. Suris.On integrability of Hirota-Kimura type discretizations, Regular Chaotic Dyn.16(2011), No. 3-4, p. 245–289

2011

[9] [9]

Petrera, J

M. Petrera, J. Smirin, Yu.B. Suris.Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. Royal Soc. A,475(2019), 20180761, 13 pp

2019

[10] [10]

Petrera, Yu.B

M. Petrera, Yu.B. Suris.New results on integrability of the Kahan-Hirota-Kimura dis- cretizations. - In:Nonlinear Systems and Their Remarkable Mathematical Structures, Ed. N. Euler, CRC Press, Boca Raton FL, 2018, p. 94–120

2018

[11] [11]

Suris.A new approach to integrals of discretizations by polarization

Yu.B. Suris.A new approach to integrals of discretizations by polarization. Open Commun. in Nonlin. Math. Phys., 2024, Special Issue in Memory of Decio Levi, 11571, 8 pp. 11

2024