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When any three solutions are independent

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

  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].

References

  1. 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)

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

    Book  Google Scholar 

  3. 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)

    MathSciNet  Article  Google Scholar 

  4. Dreyfus, T., Hardouin, C., Roques, J.: Hypertranscendence of solutions of Mahler equations. J. Eur. Math. Soc.: JEMS 20(9), 2209–2238 (2018)

    MathSciNet  Article  Google Scholar 

  5. Freitag, J., Moosa, R.: Bounding nonminimality and a conjecture of Borovik–Cherlin. J. Eur. Math. Soc. arXiv:2106.02537(to appear)

  6. Freitag, J., Scanlon, T.: Strong minimality and the \( j \)-function. J. Eur. Math. Soc. 20(1), 119–136 (2017)

    MathSciNet  Article  Google Scholar 

  7. 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)

  8. Hrushovski, E.: Almost orthogonal regular types. Ann. Pure Appl. Logic 45(2), 139–155 (1989)

    MathSciNet  Article  Google Scholar 

  9. Hrushovski, E., Itai, M.: On model complete differential fields. Trans. Am. Math. Soc. 355(11), 4267–4296 (2003)

    MathSciNet  Article  Google Scholar 

  10. Hrushovski, E., Sokolović, Ž.: Strongly minimal sets in differentially closed fields (1993) (unpublished manuscript)

  11. Jaoui, R.: Generic planar algebraic vector fields are strongly minimal and disintegrated. Algebra Number Theory 15(10), 2449–2483 (2021)

    MathSciNet  Article  Google Scholar 

  12. Jaoui, R., Jimenez, L., Pillay, A.: Relative internality and definable fibrations. arXiv:2009.06014

  13. Jin, R., Moosa, R.: Internality of logarithmic-differential pullbacks. Trans. Am. Math. Soc. 373(7), 4863–4887 (2020)

    MathSciNet  Article  Google Scholar 

  14. 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)

  15. Knop, F.: Mehrfach transitive operationen algebraischer gruppen. Arch. Math. 41(5), 438–446 (1983)

    MathSciNet  Article  Google Scholar 

  16. Kolchin, E.R.: Algebraic groups and algebraic dependence. Am. J. Math. 90, 1151–1164 (1968)

    MathSciNet  Article  Google Scholar 

  17. Kowalski, P., Pillay, A.: Quantifier elimination for algebraic \(D\)-groups. Trans. Am. Math. Soc. 358(1), 167–181 (2006)

    MathSciNet  Article  Google Scholar 

  18. 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)

  19. León Sánchez, O., Pillay, A.: Some definable Galois theory and examples. Bull. Symb. Log. 23(2), 145–159 (2017)

  20. Marker, D.: Model Theory of Differentiable Fields. Lecture Notes in Logic 5. Springer, Berlin (1996)

    Google Scholar 

  21. McGrail, T.: The search for trivial types. Ill. J. Math. 44(2), 263–271 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Mitschi, C., Singer, M.F.: Connected linear groups as differential Galois groups. J. Algebra 184(1), 333–361 (1996)

    MathSciNet  Article  Google Scholar 

  23. Moosa, R.: A nonstandard Riemann existence theorem. Trans. Am. Math. Soc. 356(5), 1781–1797 (2004)

    MathSciNet  Article  Google Scholar 

  24. Moosa, R.: On saturation and the model theory of compact Kähler manifolds. J. Reine Angew. Math. 586, 1–20 (2005)

    MathSciNet  Article  Google Scholar 

  25. 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)

  26. Moosa, R., Pillay, A.: Some model theory of fibrations and algebraic reductions. Sel. Math. New Ser. 20(4), 1067–1082 (2014)

    MathSciNet  Article  Google Scholar 

  27. Nagloo, J.: Algebraic independence of generic painlevé transcendents: PIII and PVI. Bull. Lond. Math. Soc. 52(1), 100–108 (2020)

    MathSciNet  Article  Google Scholar 

  28. Nagloo, J., Pillay, A.: On the algebraic independence of generic painlevé transcendents. Compos. Math. 150(04), 668–678 (2014)

    MathSciNet  Article  Google Scholar 

  29. Nagloo, J., Pillay, A.: On algebraic relations between solutions of a generic Painlevé equation. J. Reine Angew. Math. 726, 1–27 (2017)

    MathSciNet  Article  Google Scholar 

  30. Pila, J., Tsimerman, J.: Ax–Schanuel for the \(j\)-function. Duke Math. J. 165(13), 2587–2605 (2016)

    MathSciNet  Article  Google Scholar 

  31. Pillay, A.: Geometric Stability Theory. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  32. Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)

    MathSciNet  Article  Google Scholar 

  33. Rosenlicht, M.: The nonminimality of the differential closure. Pac. J. Math. 52, 529–537 (1974)

    MathSciNet  Article  Google Scholar 

  34. Singer, M.F.: Algebraic relations among solutions of linear differential equations: Fano’s theorem. Am. J. Math. 110(1), 115–143 (1988)

    MathSciNet  Article  Google Scholar 

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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). 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