Simple generators of rational function fields

Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and show that it improves upon the state of the art both in efficiency and the quality of the results. Furthermore, we demonstrate the utility of simplified generators through several case studies from different application domains, such as structural parameter identifiability. The main algorithmic novelties include performing only partial Gröbner basis computation via sparse interpolation and efficient search for polynomials of a fixed degree in a subfield of the rational function field.

💡 Research Summary

The paper addresses the problem of simplifying a generating set for a subfield E of the rational function field k(x₁,…,xₙ). While any such subfield can be generated by finitely many rational functions, the generators produced by existing algorithms are often extremely complicated, hindering interpretation in applications such as structural parameter identifiability. The authors propose a new algorithm that, given an initial generating set g = (g₁,…,g_m) with E = k(g), returns a “simpler” generating set h = (h₁,…,h_ℓ) where E = k(h). Simplicity is understood informally as a small number of generators, each being sparse, low‑degree, and having small coefficients.

The core technical contribution is a hybrid evaluation‑interpolation approach that avoids full Gröbner basis computation for the OMS (Oliviér‑Müller‑Quade‑Steinwandt) ideals that encode the algebraic relations of E. Instead of computing a Gröbner basis over the rational function field k(x)