"Goldfish'' equations for infinitely many particles

Francois Leyvraz

REVIEW 4 major objections 4 minor 18 references

"Goldfish'' equations for infinitely many particles

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Zeros of a simple combination of two entire functions solve the infinite-particle goldfish system.

desk verdict Promising but unfinished: Leyvraz conditional extension of the goldfish equations to infinitely many particles has a real gap at Eq. (23), and the infinite ODE may not be well-defined for legitimate examples. read the letter →

arxiv 2607.15237 v1 pith:AUCZVEUQ submitted 2026-07-16 math-ph math.MP

Francois Leyvraz This is my paper · ORCID

classification math-phmath.MP MSC 30D2034A3430C15

keywords goldfishequationsmany-bodyproblementirefunctionszerosofCalogero-typesystemsinfinite-dimensionalODEsHadamardproductRouchétheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the famous 'goldfish' many-body problem, which for N particles is exactly solvable by tracking the zeros of a polynomial, to the case of infinitely many particles. The central claim is that if p and q are entire functions of order less than 1 and q is negligible compared with p in a precise ratio sense, then the zeros of p(z) + t q(z) satisfy the infinite version of the goldfish equations. This gives a whole family of exact solutions to an infinite-dimensional nonlinear system, reducing the dynamics to the study of the zero set of a single function. A sympathetic reader would see this as a meaningful step toward understanding infinite-particle integrable systems, though the paper itself is candid that it does not describe the full dynamics or characterize which initial conditions are covered.

What carries the argument

The key machinery is the Hadamard product representation for entire functions of order less than 1: such a function is exactly an infinite product over its zeros, f(z)=∏(1 - z/z_m), with no extra exponential factors. This lets the paper treat entire functions like infinite-degree polynomials. The relation q≪p, defined by vanishing of max_{|z|=R_j}|q/p| along a sequence R_j→∞, plays the role of the polynomial-degree condition and, via Rouché's theorem, ensures that Φ(z,t) has zeros that vary continuously and analytically along a contour avoiding double zeros. The finite truncations p_N, q_N then generate finite zero sets obeying the finite goldfish equations, and the entire argument rests on

What would settle it

Take an explicit pair p, q of order less than 1 with q≪p, compute the finite-N zero sets of p_N + t q_N, and evaluate both sides of the infinite goldfish equation for a fixed index m and fixed time t. If the partial sums of the right-hand side do not converge to the second derivative of z_m(t), or if the product in equation (23) fails to converge for some m and t, the central claim is false.

Watch

Extended reading notes

Core claim

The paper proves that for two entire functions p(z) and q(z) of order less than 1, normalized by p(0)=1 and q(0)=0, satisfying q(z) ≪ p(z) (meaning |q/p| tends to 0 along a sequence of circles of growing radius), the zeros z_k(t) of Φ(z,t)=p(z)+tq(z) obey the infinite-particle goldfish equations, i.e. equation (9) with N replaced by infinity. The proof proceeds by truncating the zero sets of p and q to finite polynomials, using the known fact that finite zero sets obey the finite goldfish equations, and then passing to the limit. The crucial step is a claim about the convergence of products of ratios of zeros — equation (23) — which the paper itself describes as 'remarkably non-trivial' beca

Load-bearing premise

The proof relies on the unproved assertion that the finite truncation products in equation (23) converge to the infinite products on a common domain, including terms with indices comparable to N where the individual truncated zeros do not converge, and that this convergence is strong enough to allow differentiation.

Editorial extensions

If this is right

For any pair of entire functions p, q of order less than 1 with q≪p, the zero trajectories z_k(t) form an exact solution of the infinite goldfish equations, with initial velocities determined by the explicit formula involving q and the infinite product over the other zeros.
The condition q≪p is broad enough to include cases where q has lower order than p, or equal order but lower type, giving a large class of explicitly solvable infinite-particle dynamics.
The zeros are analytic in t along any contour in the complex t-plane that avoids the isolated times where Φ has double zeros, so solutions can be continued past algebraic singularities in the finite case.
The theorem provides only a forward map from entire functions to solutions; the paper shows that a converse reconstruction of all initial conditions is unlikely, since the interpolation problem for infinite data by an entire function has no general solution.
The framework opens a route to studying the periodized version of the goldfish equations with infinitely many particles, where the permutation of zeros after one period could be nontrivial and possibly aperiodic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The unproved convergence in equation (23) is a concrete testable point: for any explicit p and q satisfying the hypotheses, one can numerically compare the truncated product over l up to N with the infinite product for fixed m and watch whether convergence is uniform enough for the derivative exchange to hold.
Because q≪p is not a total order and can hold in both directions simultaneously, the class of admissible pairs is structurally different from the polynomial degree hierarchy; this suggests that a direct characterization of all reachable initial conditions may require a finer invariant than order or type.
If the same machinery could be extended to entire functions of order ρ≥1 with Hadamard exponential factors, the zero sets would no longer have a unique simple product representation, so the goldfish equations would likely need modification; this is a natural but unexplored boundary.
The author's speculation about chaotic behavior in the infinite-periodic case is not established here, but the framework gives a concrete way to test it: choose a periodic entire Φ(z,t), track the induced permutation on infinitely many zeros, and examine whether finite truncations show non-periodic or sensitive dependence on initial conditions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

If you track Calogero-type systems, this paper is worth a look. Leyvraz takes the classic finite-N goldfish solution (zeroes of p(z)+tq(z) move according to a simple ODE) and tries to push it to N=∞ by replacing polynomials with entire functions of order <1. The new piece is the q≪p condition as an analog of degree, plus a useful singular example (cos√z) showing what can go wrong. The paper is transparent: it explicitly says the converse is open, the dynamics are not characterized, and it does not claim more than a conditional result. That honesty counts for something.

What the paper does well: the finite-N normalization and initial-value formulas are clean and correct. The q≪p idea is plausible and the singular example is genuinely instructive. The proof strategy — approximate by finite truncations and then pass to the limit — is natural.

Where it gets soft: the central step is Eq. (23), where the product over truncated zeroes is asserted to converge to the infinite product. The paper itself calls this "remarkably non-trivial" and gives no proof. That is a genuine gap, not a minor annoyance. Even if that limit holds, the right-hand side of Eq. (9) is an infinite series whose convergence is never established. A stress-test example p(z)=cos√z, q(z)=z satisfies q≪p, but the terms in the sum behave like 1/k, so the series fails absolute convergence. That means the infinite ODE is not unambiguously defined unless an ordering convention is specified. The paper does not address this. This is a load-bearing flaw, not a technicality.

That said, the paper does not overclaim. It is an exploratory note that identifies the hard analytic problems rather than hiding them. A referee could reasonably ask for a rigorous treatment of Eq. (23) and for a definition of what the infinite sum means. The idea is novel enough and the limitations are candid enough that it deserves a serious referee. The kind of reader who gets value is someone working on integrable many-body systems or entire functions, and willing to work on the open problems. I would not cite it in my next paper, but I would bring it to a reading group.

Recommendation: send it to peer review. It will need heavy revision, but there is a real idea here and the author's honesty gives a solid starting point.

Referee Report

4 major / 4 minor

Summary. The paper proposes an infinite-particle extension of the Calogero "goldfish" equations. For polynomials, the zeroes of Phi(z,t)=p(z)+tq(z) satisfy the finite-N system (1) (or (9) under the normalization p(0)=1, q(0)=0). The author considers entire functions p,q of order rho<1 with p(0)=1, q(0)=0, introduces a dominance condition q<<p expressed by decay of |q/p| along a sequence of circles, and claims that the zeroes z_k(t) of Phi(z,t)=p(z)+tq(z) satisfy Eq. (9) with N replaced by infinity, with initial velocities given by (26). The strategy is to truncate the zero products to finite N, use the finite-N goldfish equations, and then pass to the limit; regularity of the zeroes is addressed via Rouche's theorem and a contour argument. The paper is explicit that the converse problem--characterizing the dynamics and the set of attainable initial conditions--remains open.

Significance. The finite-N portion is standard and correct, and the paper contains useful observations: the example in Sec. IIA shows that order alone does not replace the notion of degree, and the condition q<<p is a clean, parameter-free sufficient condition for the zeroes to remain regular. If the infinite-dimensional claim could be rigorously established, it would provide an exact correspondence between zeroes of a deforming entire function and an infinite many-body system, which is a potentially valuable contribution to the solvable-dynamics literature. However, the central limit step is not proved, and the stated infinite ODE is not shown to be well-defined. The paper's own concluding remarks appropriately acknowledge that the dynamics is not characterized, but the abstract and Sec. III state the main theorem more strongly than the proof supports.

major comments (4)

[II.B, Eq. (23)] The convergence (23) is the key step transferring the finite-N goldfish equations to N=infinity, but it is only asserted as "remarkably non-trivial." For l comparable to N, z_l^{(N)}(t) need not converge to z_l(t), so (23) is not a consequence of pointwise convergence of the roots. A rigorous proof is required, for example via uniform estimates from Rouche's theorem or by a direct comparison of the canonical products. Without it, the limit N->infinity is not justified.
[II.B, Eqs. (21)-(23) and the following paragraph] Even if (23) is granted, the passage from (23) to the statement that the z_m(t) satisfy (9) with N=infinity is incomplete. The paper differentiates the convergence statement without proving uniform convergence of the derivatives, and Eq. (21a) itself is an infinite-series identity obtained by differentiating an infinite product. The required interchanges of limits and derivatives are not established. A precise convergence mode (for example, locally uniform in t) and a proof are needed.
[II.B / III, Eq. (9) with N=infinity] The right-hand side of (9) is never defined as a convergent infinite series. For p=cos(sqrt(z)), q=z, which satisfy the hypotheses (order 1/2 and 0, q<<p), the k-th term at t=0 has magnitude ~ 2 sqrt(z_k)/|z_m-z_k| ~ (2/pi) k/|m^2-k^2|, so the series is not absolutely convergent. It may be conditionally convergent for this example, but no summation convention is specified and no general convergence proof is supplied. The claimed theorem is therefore not yet a statement about a well-defined infinite-dimensional ODE.
[II.C / III] The theorem in Sec. III states that q<<p suffices, but Sec. II.C only establishes continuity and local analyticity of each z_m(t). It does not supply the missing analytic estimates needed for Eqs. (21)-(23). In addition, the assertion that the double zeroes of Phi are isolated because the number of z_alpha^{(N)} in every disk remains bounded is not proved. At minimum, the hypotheses actually used in the proof should be stated as explicit assumptions and distinguished from the sufficient condition q<<p.

minor comments (4)

[I, Eq. (2)] The text has "p(x)" in the definition of Phi(z,t); this should be "p(z)".
[II.B] The notation "Phi_N(z.t)" and "p_N(t)" should be "Phi_N(z,t)" and "p_N(z)", respectively.
[III] The sentence "We have shown that ..." overstates the conditional nature of the proof; recommend qualifying with the hypotheses and convergence assumptions used in Sec. II.B.
[III, Eq. (26b)] The initial-velocity formula would benefit from a derivation or a reference; the inverse-product factor is obtained from (22a), but the notation is not immediately transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the infinite-particle goldfish claim is derived from Hadamard factorization plus an unproved convergence interchange; the flagged gap is a correctness matter, not an input–output identity.

full rationale

The derivation chain is not circular. The finite-N goldfish equations (9) are standard consequences of the polynomial factorization (8); the paper then lifts them to N=infinity by defining z_k(t) through the infinite product (20), introducing finite truncations Phi_N, and passing to limits. The central step is the convergence statement (23), which the paper itself flags as “a remarkably non-trivial relation” since “no convergence, nor indeed any control, can be straightforwardly shown” for z_l^(N) with l comparable to N. That is an omitted proof / missing support, not a circular reduction: no parameter is fitted to the predicted ODE, no normalization is chosen so that (9) holds by construction, and no self-citation supplies the load-bearing conclusion. The only self-citation, [5], is contextual background on the Hamiltonian character of the finite system and is not used to establish the infinite-particle statement. The conclusion itself is explicitly modest (“we have not characterized this dynamics at all, nor shown that it exists”), and the skeptic’s concern about non-absolute convergence of the RHS of (9) (e.g., for p=cos(sqrt z), q=z) is a well-definedness issue for the infinite ODE, again independent of circularity. Thus the paper’s claims reduce neither by definition nor by self-citation to their inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical fitting constants, no new physical entities. The only inputs are the functions p and q and technical regularity conditions.

assumptions (4)

standard math Entire functions of order ρ<1 with f(0)=1 have a unique absolutely convergent product representation f(z)=∏(1-z/z_n) with no exponential prefactor (Hadamard factorization, genus 0).
Used to pass from zero sets to functions; §II, Eq. (15) relies on Hadamard theory.
domain assumption There exists an open neighborhood G of the real interval/contour C on which every zero trajectory z_n(t) is analytic, and a sequence of circles R_j with max_{|z|=R_j}|q/p|→0 (q≪p).
The theorem is conditional on these regularity hypotheses; introduced in §II.B and §II.C, Eq. (25).
standard math Rouché's theorem and the implicit function theorem guarantee continuous/analytic zero branches away from double zeros, and double-zero times are isolated.
Used in §II.C to build the contour C; assumes no accumulation of double zeros along C.
ad hoc to paper The finite-N truncated products in Eq. (23) converge uniformly on a common domain to the infinite product, and derivatives may be interchanged with the limit.
This is the load-bearing unproved exchange; stated as 'remarkably non-trivial' in §II.B and then used without proof.

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Cite this review

Pith. "Pith review of "Goldfish'' equations for infinitely many particles." pith.science (2026). https://pith.science/paper/AUCZVEUQ

@misc{pith2026260715237,
  author       = {Pith},
  title        = {Pith review of: "Goldfish'' equations for infinitely many particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUCZVEUQ}},
  note         = {Machine review of arXiv:2607.15237}
}

abstract

The ``goldfish'' equations, so named because of their striking beauty, are a system of $N$ nonlinear ODE's, where $N$ is an arbitrary integer. They can be solved exactly in a very simple manner, by transforming them to free motion using a transformation involving the transition from the set of {\em coefficients\/} of a polynomial to that of its {\em zeroes}. This paper aims to explore the possibility of extending this solution to the case in which $N$ is infinite. The main difficulty involves the transition from polynomials to entire functions. Another approach using non-standard analysis, is left to future work.

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Works this paper leans on

18 extracted references

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Solvable

F. Calogero. Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations and Related “Solvable” Many-Body Problems. Il Nuovo Cimento43 B, (2) 177–241 (1978) 13

1978

[2] [2]

goldfish

F. Calogero. The neatest many-body problem amenable to exact treatments (a “goldfish”?). Physica D152–153(2001) 78–84

2001

[3] [3]

Calogero and J.-P

F. Calogero and J.-P. Fran¸ coise. Hamiltonian character of the motion of the zeros of a poly- nomial whose coefficients oscillate over time. J. Phys. A: Math. Gen.30(1997) 211–218

1997

[4] [4]

M.C. Nucci. 2004. Calogero’s ‘goldfish’ is indeed a school of free particles. Journal of Physics A: Mathematical and General, 37 (47), pp. 11391–11400

2004

[5] [5]

F. Leyvraz. An approach for obtaining integrable Hamiltonians from Poisson-commuting poly- nomial families. J. Math. Phys.58, 072902 (2017)

2017

[6] [6]

Calogero

F. Calogero. A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic. J. Math. Phys.38, 5711 (1997)

1997

[7] [7]

G´ omez-Ullate and M

D. G´ omez-Ullate and M. Sommacal. 2005. Periods of the goldfish many-body problem. Journal of Nonlinear Mathematical Physics, 12 (suppl.), pp. 351–362

2005

[8] [8]

Calogero

F. Calogero. Solution of the GoldfishN-Body Problem in the Plane with (only) Nearest- Neighbor Coupling Constants all equal to minus one half. Journal of Nonlinear Mathematical Physics Volume11, (1) (2004), 102–112

2004