Restrictions for General Algebraic Integrability

Determine whether arbitrary derivations over a number field with finitely many variables satisfy additional algebraic or arithmetic restrictions analogous to the degree-at-most-two residue-field restriction proved for separated-variable vector fields, in order to advance the general algebraic integrability conjecture.

Background

The paper proves that, in a particular two-variable separated-variable setting where the partial fraction decompositions of the reciprocals contain only logarithmic parts, algebraic integrability forces the field generated by the residues to have degree at most two over the base field. This yields strong explicit constraints on the numerators and supports the paper’s formulas over the rational numbers.

The authors then ask whether comparable restrictions exist for arbitrary derivations over number fields with finitely many variables. Such restrictions could provide arithmetic tools for approaching the unresolved general Algebraic Integrability Conjecture beyond the special separated-variable class treated in the paper.

References

For arbitrary derivation $D$ on a field over a number field with finitely many variables. Are there more complex restrictions similar to Corollary \ref{cor: degree no more than 2} that we could exploit towards the general algebraic integrability conjecture \ref{conj: alg int conj}?

— On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables  (2609.00974 - Grabowski, 1 Sep 2026) in Section 5, immediately after Corollary \ref{cor: degree no more than 2}, Problem “Algebraic Number Theory meets Vector Fields”