Some functional transcendence results around the Schwarzian differential equation.

Abstract

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Strong minimality of triangle functions

“…In this section, we use the results of Section 3 with the model theoretic notions of Section 2 to give a new proof of the strong minimality of the equations satisfied by Schwarz triangle functions (and so including the j-function). In particular these results strengthen the main theorems of [19,18], generalize the main theorem of [9], and give new proofs of results of [2,3] in the case of Schwarz triangle functions. Proof.…”

“…This classification of algebraic solutions to 3.1 applies more generally to the case where α, β, γ ∈ C ∪ {∞} and comes ultimately from Theorem I of [12], but is explained in detail in [2,Proposition 4.4].…”

“…This generalization is equivalent to the strong minimality of the equation, a notion having its origins in model theory. In this short article we give a new proof of the strong minimality of Schwarz triangle functions (originally results of [9,2], later [1]) using the recent work of [8].…”

On the density of strongly minimal algebraic vector fields

“…x 3 ) for well-chosen values of α 1 , α 2 , α 3 ∈ C 3 are strongly minimal and geometrically trivial. This follows from the results of [BSCFN20] and was kindly pointed out to the author by Ronnie Nagloo.…”

“…In the setting of differential algebra, the effectiveness of these model-theoretic techniques to produce transcendence results has been demonstrated both from a practical and from a theoretical perspective. 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].…”