Generic planar algebraic vector fields are strongly minimal and disintegrated

On the density of strongly minimal algebraic vector fields

“…On the other hand, in dimension two 9 , the hypothesis (a) is not necessary. This follows from Theorem C together with the results of [Jao21]. In dimension ≥ 4, this assumption will be needed in our analysis in the form of Corollary 4.2.6.…”

“…Under the assumptions (a) and (b), we are able to apply the results of [Per12] to conclude that the canonical invariant hypersurface is the only horizontal subvariety of type (I). The horizontal subvarieties of type (II) are studied in subsections 4.3 and 4.4 using an argument -based on the study of the tangency locus of an invariant distribution of codimension one with the foliation F (v)similar to the one used in dimension two in [Jao21].…”

“…To handle the case where dim(X) = 2, it is enough to see that (b) ensures that every rational integral of (X, v) is constant. This follows from a theorem of Camacho and Sad [CS80] which ensures that every singular point admits at least one analytic separatrix (see the proof of Theorem 2.1.2 in [Jao21] for details).…”

“…For instance, after constructing an example satisfying (1) -the Poizat equation y ′′ • y = y ′ and y ′ = 0 -, Poizat conjectures in [Poi83] that for every n ≥ 2, a "general" order n algebraic (non linear) differential equation should be strongly minimal (see also Conjecture 1 in [She73]) and concludes that: à propos des corps differentiels, on est souvent amené à faire des conjectures; dont on est persuadé qu'elles ne peuvent être fausses que pour des équations très particulières, et que pourtant on n'arrive à montrer que dans des cas encore plus particuliers. 1 After the many progress concerning classical families mentioned above, the first abundance result for strongly minimal differential equations of order two obtained in [Jao21] states that the family Ξ(2, d) formed by the systems of differential equations of the form (S) :…”

“…Statement of the main results. In this article, we generalize the techniques developed in [Jao21] in dimension two to arbitrary dimension and produce two abundance results for families of autonomous algebraic differential equations satisfying properties (1) and (2). The first concerns the systems of polynomial differential equations on the affine n-space.…”