When any three solutions are independent

On the density of strongly minimal algebraic vector fields

“…It is organized as follows. In subsection 2.1, we recall the correspondence between algebraic vector fields and their solution sets in a differentially closed field for which there are many references in the model-theoretic literature (see for example the first section of [HI03] or [FJM22]). The subsections 2.2 to 2.4 introduce some vocabulary on twisted algebraic vector fields that will be used in Section 5.…”

“…On the practical side, they led in various classical families of algebraic differential equations -such as the families of Painlevé equations [NP14] [NP17] [FN22], families of Schwarzian equations [FS18] [CFN20] [BSCFN20] and certain families of Liénard equations [FJMN22] -to a full understanding of the algebraic relations between solutions of equations in the family under study. From a theoretical perspective, they provided new insights on the possible algebraic relations in an arbitrary family of algebraic differential equations [FJM22].…”

A Classification of First Order Differential Equations

“…In a recent article [4], Freitag et al study the algebraic dependence of solutions of differential equations of any order using model theoretic methods. They show that if a first order differential equation (respectively, an autonomous differential equation) has four (respectively, two) algebraically independent solutions then any m distinct nonalgebraic solutions are algebraically independent.…”

A model theory for meromorphic vector fields

“…(1) to show that the (universal) theory of differential CCM-structures admits a model companion, which is DCCM, by giving a geometric first-order axiomatisation of the existentially closed models (Theorem 5.5); (2) to show that DCCM is complete, admits quantifier elimination (Proposition 6.3) and elimination of imagainaries (Theorem 7.6), and to give a geometric characterisation of definable and algebraic closure (Proposition 6.4); (3) to show that DCCM is totally transcendental (Theorem 7.5), and to give geometric characterisations of nonforking independence (Corollary 7.4); and, (4) to establish the correspondence between finite-dimensional types (over the empty set) in DCCM and meromorphic vector fields (Theorem 8.3). The proofs proceed largely by finding geometric analogues for the algebraic arguments already familiar from DCF 0 .…”

“…In any case, once the canonical base property is established, a concrete manifestation of the Zilber dichotomy for finite dimensional minimal types in DCCM will follow. It would then be reasonable to expect that many of the recent applications of model theory to algebraic vector fields, as carried out in [4] and [6] for example, would extend to meromorphic vector fields.…”