I will discuss algebraic transcendence properties — and, more specifically, the property of disintegration — for the solutions of planar complex polynomial vector fields.
This fundamental property, originating from the work of Zilber on strongly minimal sets, expresses that any algebraic relation between solutions of the differential equation under study can be decomposed as an intersection of algebraic relations between only two solutions.
I will explain a recent result that I proved concerning the "very generic case of degree d" where all the coefficients of the polynomial vector field are Q-algebraically independent. Then, I will discuss some possible strengthening of this result and speculate on the possible kinds of disintegration phenomena that it is possible to encounter for planar algebraic vector fields.