Further Structural Use of m-Saturation

Determine whether the m-saturation of a foliation—defined using membership in sums of its iterated \(p\)-powers modulo almost all primes—has further mathematical use, including whether it admits structural propositions beyond the separated-variable example analyzed in the paper.

Background

The paper introduces m-saturation to capture derivations that lie, modulo almost all primes, in the span of a foliation and its iterated pp-powers. It shows that this construction is itself a foliation and computes it explicitly for separated-variable derivations by organizing the coefficients into several residue and constant-type classes.

The construction is presented as a possible tool for studying rational first integrals in arbitrary dimension. However, the paper provides only one principal example and explicitly states that its broader usefulness and structural theory remain unresolved.

References

We do not know if it is of any further use. To judge it properly, one would require a more diverse set of examples and some structural propositions about this notion.

— On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables  (2609.00974 - Grabowski, 1 Sep 2026) in Section 4.1, paragraph immediately preceding Example \ref{Ex: Computing a m-saturation}