Discrete Integrable Equations over Finite Fields

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.

💡 Research Summary

This paper tackles the longstanding difficulty of defining and solving discrete integrable equations directly over finite fields, where the usual algebraic operations can lead to indeterminate forms such as 0/0. The authors propose a systematic regularisation technique: instead of working in the finite field (\mathbb{F}_p) itself, they lift the problem to the field of rational functions (\mathbb{F}_p(\epsilon)), where (\epsilon) is an auxiliary infinitesimal parameter. By expanding all quantities as polynomials in (\epsilon), performing the algebraic manipulations, and finally taking the limit (\epsilon\to0), one obtains well‑defined expressions that reduce back to the original finite field. This approach preserves the discrete nature of the underlying arithmetic while eliminating the singularities that obstruct direct computation.

The central object of study is a generalized discrete Korteweg–de Vries (KdV) equation of the form
\