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When any three solutions are independent


Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. In the autonomous situation when the equation is over constant parameters the assumption that the order be greater than one can be dropped, and a nontrivial algebraic relation exists already between two solutions. These theorems are deduced as an application of the following model-theoretic result: Suppose p is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of p are independent then p is minimal. If the type is over the constants then minimality (and complete disintegratedness) already follow from knowing that any two realisations are independent. An algebro-geometric formulation in terms of D-varieties is given. The same methods yield also an analogous statement about families of compact Kähler manifolds.

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  1. More generally our results apply to compact complex analytic varieties in Fujiki’s class \({\mathscr {C}}\). In fact, all that is needed is that they be essentially saturated in the sense of [24].


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Correspondence to Rahim Moosa.

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The first author was partially supported by NSF grant DMS-1700095 and NSF CAREER award 1945251. The second author was partially supported by the ANR-DFG program GeoMod (Project number 2100310201). The third author was partially supported by an NSERC Discovery Grant. The first and third authors would also like to thank the hospitality of the Fields Institute in Toronto during the 2021 thematic programme on Trends in Pure and Applied Model Theory, where much of this work was carried out.

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Freitag, J., Jaoui, R. & Moosa, R. When any three solutions are independent. Invent. math. (2022).

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Mathematics Subject Classification

  • 03C45
  • 12H05
  • 11J81
  • 32J27