This paper asks whether a neat trick that solves a famous finite system of equations can be pushed to the infinite case. The original “goldfish” equations are a system of N coupled nonlinear ordinary differential equations (ODEs) for N complex variables. They can be solved exactly by turning the problem of particle positions into the problem of the zeros of a polynomial p(z)+t q(z). That transform turns the interacting motion into free motion of the polynomial coefficients, which is easy to solve.

The author tries to carry this method over when N goes to infinity. Polynomials are replaced by entire functions — functions that are analytic everywhere in the complex plane. To keep the same normalization used in the finite case, the paper considers entire functions p and q with p(0)=1 and q(0)=0, and studies the zeros zk(t) of Φ(z,t)=p(z)+t q(z). For a large class of entire functions that grow slowly (those of order ρ<1, meaning roughly that they grow like exp(|z|^ρ) with ρ<1), classical results by Hadamard give a product representation p(z)=∏(1−z/zn). That representation makes the infinite case look formally similar to the polynomial case and gives hope the trick might work.

The paper explains two main obstacles. First, the replacement of degree (a finite integer for polynomials) by growth order for entire functions is not a perfect substitute. The author gives a concrete example built from cosine of √z that shows some zeros can blow up as t→0 while others stay near their initial values. In that example some families of zeros diverge like −1/(2t) or as −(ln t)^2 when t is small. Those diverging zeros have no relation to the initial zeros and show a behavior analogous to mismatched polynomial degrees in the finite case.

Second, the passage from finite to infinite brings analytic and convergence issues. Mapping initial particle positions and velocities to p and q becomes an interpolation problem for infinitely many values. Existence and uniqueness of an entire interpolating function are not guaranteed. The implicit function theorem, which guarantees smooth dependence of zeros on t when zeros stay simple, becomes harder to use because entire functions can have sequences of pairs of zeros that come arbitrarily close together. The author therefore works under additional regularity assumptions: the zeros zk(t) are continuous on [0,T] and analytic in a complex neighborhood of the t-path, and double zeros are avoided along that path.