Algebraicity and minimal polynomial of the reduced hierarchy

Prove algebraicity of the distinguished reduced generating function for arbitrary recursion order p and construct its minimal polynomial, whose conjectured degree is 2(p−1).

Background

The paper establishes exact algebraic equations for the reduced hierarchy at p=4 and p=5, together with an exact p=6 specialization, exhibiting degrees 6, 8, and 10. These computations motivate the conjectured general degree 2(p−1) and the proposed global 2(p−1)-sheeted covering.

The authors explicitly identify the extension from these computed cases to arbitrary p as unresolved. In particular, they state that the boundary factorization suggests the covering but that a direct construction of the minimal polynomial is not currently known.

References

Several problems remain. The first is to prove algebraicity of the reduced hierarchy for arbitrary $p$. The boundary factorization suggests a $2(p-1)$-sheeted global covering, but a direct construction of its minimal polynomial is not yet known.

— Differential Recursions, Projective Discriminants, and Algebraic Generating Functions  (2609.31189 - Voloshyn, 25 Sep 2026) in Section 7, “General conjectures and open problems” (Section \ref{sec:conjectures})