Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

TL;DR

This paper investigates whether the susceptibility of the square-lattice Ising model is a differentially algebraic function, exploring its properties modulo primes and providing extensive series data to analyze its algebraic nature.

Contribution

The study extends the series for the Ising model susceptibility to 5043 coefficients and examines its reduction modulo various primes, offering new insights into its algebraic and differential properties.

Findings

Series reduces to algebraic functions modulo powers of 2.

Long series insufficient to determine algebraic reduction modulo 3, 5, etc.

Examples of integer coefficient series reduce to algebraic functions modulo almost all primes.

Abstract

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation. Initial results on Tutte's non-linear ordinary differential equation (ODE) and other simple quadratic non-linear ODEs suggest that a large set of differentially algebraic power series solutions with integer coefficients might reduce to algebraic functions modulo primes, or powers of primes. Here we give several examples of series with integer coefficients and non-zero radius of convergence that reduce to algebraic functions modulo (almost) every prime (or power of a prime). These examples satisfy differentially algebraic equations with the encoding polynomial occasionally possessing quite high degree (and thus difficult to identify even with long series). Additionally, we have extended both the high- and low-temperature Ising square-lattice susceptibility series to 5043 coefficients. We find that even this long series is insufficient to determine whether it reduces to algebraic functions modulo $3$, $5$, etc. This negative result is in contrast to the comparatively easy confirmation that the corresponding series reduce to algebraic functions modulo powers of $2$.