On the density of strongly minimal algebraic vector fields

Abstract : Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree \( d > 1 \) on the affine space of dimension \( n > 1 \) is strongly minimal and geometrically trivial. The second one states that if \(X_0\) is the complement of a smooth hyperplane section \( H_X \) of a smooth projective variety \( X \) of dimension \( n > 1 \) then for \(d \) sufficiently large, the system of differential equations associated with a generic vector field on \( X_0 \) with poles of order at most \( d \) along \( H_X \) is also strongly minimal and geometrically trivial.