## Abstract

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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## Notes

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].

## References

Borovik, A., Cherlin, G.: Permutation groups of finite Morley rank. In: Model Theory with Applications to Algebra and Analysis, Vol. 2, volume 350 of London Mathematical Society Lecture Note Series, pp. 59–124. Cambridge Univ. Press, Cambridge (2008)

Buium, A.: Differential Algebraic Groups of Finite Dimension: Lecture Notes in Mathematics 1506. Springer, Berlin (1992)

Casale, G., Freitag, J., Nagloo, J.: Ax–Lindemann–Weierstrass with derivatives and the genus 0 Fuchsian groups. Ann. Math. (2)

**192**(3), 721–765 (2020)Dreyfus, T., Hardouin, C., Roques, J.: Hypertranscendence of solutions of Mahler equations. J. Eur. Math. Soc.: JEMS

**20**(9), 2209–2238 (2018)Freitag, J., Moosa, R.: Bounding nonminimality and a conjecture of Borovik–Cherlin. J. Eur. Math. Soc. arXiv:2106.02537

**(to appear)**Freitag, J., Scanlon, T.: Strong minimality and the \( j \)-function. J. Eur. Math. Soc.

**20**(1), 119–136 (2017)Fujiki, A.: On the structure of compact complex manifolds in C. In: Algebraic Varieties and Analytic Varieties (Tokyo, 1981), vol. 1 of Advanced Studies in Pure Mathematics, pp. 231–302. North-Holland, Amsterdam (1983)

Hrushovski, E.: Almost orthogonal regular types. Ann. Pure Appl. Logic

**45**(2), 139–155 (1989)Hrushovski, E., Itai, M.: On model complete differential fields. Trans. Am. Math. Soc.

**355**(11), 4267–4296 (2003)Hrushovski, E., Sokolović, Ž.: Strongly minimal sets in differentially closed fields (1993)

**(unpublished manuscript)**Jaoui, R.: Generic planar algebraic vector fields are strongly minimal and disintegrated. Algebra Number Theory

**15**(10), 2449–2483 (2021)Jaoui, R., Jimenez, L., Pillay, A.: Relative internality and definable fibrations. arXiv:2009.06014

Jin, R., Moosa, R.: Internality of logarithmic-differential pullbacks. Trans. Am. Math. Soc.

**373**(7), 4863–4887 (2020)Klingler, B., Ullmo, E., Yafaev, A.: Bi-algebraic geometry and the André–Oort conjecture. In: Algebraic Geometry: Salt Lake City 2015, volume 97 of Proceedings of Symposia in Pure Mathematics, pp. 319–359. Amer. Math. Soc., Providence (2018)

Knop, F.: Mehrfach transitive operationen algebraischer gruppen. Arch. Math.

**41**(5), 438–446 (1983)Kolchin, E.R.: Algebraic groups and algebraic dependence. Am. J. Math.

**90**, 1151–1164 (1968)Kowalski, P., Pillay, A.: Quantifier elimination for algebraic \(D\)-groups. Trans. Am. Math. Soc.

**358**(1), 167–181 (2006)León Sánchez, O., Moosa, R.: Isolated types of finite rank: an abstract Dixmier–Moeglin equivalence. Selecta Math. (N.S.)

**25**(1):Paper No. 10, 10 (2019)León Sánchez, O., Pillay, A.: Some definable Galois theory and examples. Bull. Symb. Log.

**23**(2), 145–159 (2017)Marker, D.: Model Theory of Differentiable Fields. Lecture Notes in Logic 5. Springer, Berlin (1996)

McGrail, T.: The search for trivial types. Ill. J. Math.

**44**(2), 263–271 (2000)Mitschi, C., Singer, M.F.: Connected linear groups as differential Galois groups. J. Algebra

**184**(1), 333–361 (1996)Moosa, R.: A nonstandard Riemann existence theorem. Trans. Am. Math. Soc.

**356**(5), 1781–1797 (2004)Moosa, R.: On saturation and the model theory of compact Kähler manifolds. J. Reine Angew. Math.

**586**, 1–20 (2005)Moosa, R., Pillay, A.: Model theory and Kähler geometry. In: Model Theory with Applications to Algebra and Analysis. Vol. 1, volume 349 of London Mathematical Society Lecture Note series, pp. 167–195. Cambridge Univ. Press, Cambridge (2008)

Moosa, R., Pillay, A.: Some model theory of fibrations and algebraic reductions. Sel. Math. New Ser.

**20**(4), 1067–1082 (2014)Nagloo, J.: Algebraic independence of generic painlevé transcendents: PIII and PVI. Bull. Lond. Math. Soc.

**52**(1), 100–108 (2020)Nagloo, J., Pillay, A.: On the algebraic independence of generic painlevé transcendents. Compos. Math.

**150**(04), 668–678 (2014)Nagloo, J., Pillay, A.: On algebraic relations between solutions of a generic Painlevé equation. J. Reine Angew. Math.

**726**, 1–27 (2017)Pila, J., Tsimerman, J.: Ax–Schanuel for the \(j\)-function. Duke Math. J.

**165**(13), 2587–2605 (2016)Pillay, A.: Geometric Stability Theory. Oxford University Press, Oxford (1996)

Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math.

**78**, 401–443 (1956)Rosenlicht, M.: The nonminimality of the differential closure. Pac. J. Math.

**52**, 529–537 (1974)Singer, M.F.: Algebraic relations among solutions of linear differential equations: Fano’s theorem. Am. J. Math.

**110**(1), 115–143 (1988)

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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). https://doi.org/10.1007/s00222-022-01143-8

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DOI: https://doi.org/10.1007/s00222-022-01143-8

### Mathematics Subject Classification

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