Generalizing the n+2 bound for algebraic dependence of solutions to all orders

Determine whether, for every positive integer n, the following equivalence holds for irreducible algebraic differential equations f(t,t',...,t^{(n)})=0 over an algebraically closed differential field k with constants C and any nonalgebraic solution y whose field of constants is C: the set S_{k,f,y} of solutions y1 in a universal differential extension satisfying (i) there is a k-differential isomorphism from k⟨y⟩ to k⟨y1⟩ sending y to y1, (ii) for all finite tuples y1,...,ym in S_{k,f,y} one has C_{k⟨y1,...,ym⟩}=C, and (iii) tr.deg(k⟨y1,...,ym⟩|k)=Σ_i tr.deg(k⟨yi⟩|k), is finite if and only if S_{k,f,y} has at most n+2 elements.

Background

The paper proves, for algebraic differential equations of order n ≤ 4, that the set S_{k,f,y} of solutions equivalent to a given nonalgebraic solution y (in a precise differential-algebraic sense) is finite if and only if it has at most n+2 elements.

Their proof relies on classifications of linear algebraic groups having affine homogeneous spaces of dimension ≤ 4, necessitating the restriction n ≤ 4. Extending beyond this dimension requires new methods or classifications.

The authors explicitly acknowledge that it is unknown whether the same equivalence remains true for arbitrary order n, thus posing a concrete open question about generalizing the n+2 bound to all orders.