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Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

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Abstract

This paper deals with the quantitative Schmidt’s subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt’s subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.

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References

  1. Cartan, H.: Sur les zéroes des combinaisons linéaries de p fonctions holomorphes données. Mathematica 7, 80–103 (1933)

    Google Scholar 

  2. Chen, Z., Ru, M., Yan, Q.: The truncated second main theorem and uniqueness theorems. Sci. China Math. 53, 605–616 (2010)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z., Ru, M., Yan, Q.: The degenerated second main theorem and Schmidt’s subspace theorem. Sci. China Math. 55, 1367–1380 (2012)

  4. Corvaja, P., Zannier, U.: On a general Thue’s equation. Am. J. Math. 126, 1033–1055 (2004)

  5. Evertse, J.H., Ferretti, R.G.: Diophantine inequalities on projective varieties. Int. Math. Res. Not. 25, 1295–1330 (2002)

    Article  MathSciNet  Google Scholar 

  6. Evertse, J.H., Ferretti, R.G.: A generalization of the subspace theorem with polynomials of higher degree. In: Tichy, R.F., Schlickewei, H.P., Schmidt, K. (eds.) Diophantine Approximation, Festschrift for Wolfgang Schmidt, pp. 175–198. Springer, Vienna (2008)

    Chapter  Google Scholar 

  7. Evertse, J.H., Schlickewei, H.P.: A quantitative version of the absolute subspace theorem. J. Reine Angew. Math. 548, 21–127 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. II. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  9. Giang, L.: On the quantitative subspace theorem. J. Number Theory 145, 474–495 (2014)

    Article  MathSciNet  Google Scholar 

  10. Giang, L.: An explicit estimate on multiplicity truncation in the degenerated second main theorem. Houst. J. Math. 42, 447–462 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Lei, S., Ru, M.: An improvement of Chen-Ru-Yan’s degenerated second main theorem. Sci. China Math. 58, 2517–2530 (2015)

  12. Nochka, E.I.: On the theory of meromorphic functions. Sov. Math. Dokl. 27, 377–381 (1983)

    MATH  Google Scholar 

  13. Noguchi, J.: A note on entire pseudo-holomorphic curves and the proof of Cartan–Nochka’s theorem. Kodai Math. J. 28, 336–346 (2005)

  14. Osgood, C.F.: A number theoretic-differential equations approach to generalizing Nevanlinna theory. Indian J. Math. 23, 1–15 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Osgood, C.F.: Sometimes effective Thue–Siegel–Roth–Schmidt–Nevanlinna bounds, or better. J. Number Theory 21, 347–389 (1985)

    Article  MathSciNet  Google Scholar 

  16. Quang, S.D.: Schmidt’s subspace theorem for moving hypersurfaces in subgeneral position. Int. J. Number Theory 14(1), 103–121 (2018)

  17. Quang, S.D.: A generalization of the subspace theorem for higher degree polynomials in subgeneral position. Int. J. Number Theory 15(4), 775–788 (2019)

    Article  MathSciNet  Google Scholar 

  18. Quang, S.D.: Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces. Trans. Am. Math. Soc. 371(4), 2431–2453 (2019)

    Article  MathSciNet  Google Scholar 

  19. Ru, M., Stoll, W.: The second main theorem for moving targets. J. Geom. Anal. 1, 99–138 (1991)

    Article  MathSciNet  Google Scholar 

  20. Ru, M., Stoll, W.: The Cartan conjecture for moving targets. In: Proceedings of Symposia in Pure Mathematics, vol. 52, Part 2, pp. 477–508. American Mathematical Society (1991)

  21. Ru, M.: On a general form of the second main theorem. Trans. Am. Math. Soc. 349, 5093–5105 (1997)

    Article  MathSciNet  Google Scholar 

  22. Ru, M.: A defect relation for holomorphic curves intersecting hypersurfaces. Am. J. Math. 126, 215–226 (2004)

    Article  MathSciNet  Google Scholar 

  23. Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. 169, 255–267 (2009)

    Article  MathSciNet  Google Scholar 

  24. Schmidt, W.M.: Norm form equations. Ann. Math. 96, 526–551 (1972)

    Article  MathSciNet  Google Scholar 

  25. Schmidt, W.M.: Simultaneous approximation to algebraic numbers by elements of a number field. Monatsh. Math. 79, 55–66 (1975)

    Article  MathSciNet  Google Scholar 

  26. Schmidt, W.M.: Diophantine Approximation. Lecture Notes in Mathematics, vol. 785. Springer, Berlin (1980)

    MATH  Google Scholar 

  27. Schmidt, W.M.: The subspace theorem in Diophantine approximation. Compos. Math. 96, 121–173 (1989)

    MathSciNet  MATH  Google Scholar 

  28. Shirosaki, M.: Another proof of the defect relation for moving targets. Tôhoku Math. J. 43, 355–360 (1991)

    Article  MathSciNet  Google Scholar 

  29. Vojta, P.: Diophantine Approximation and Value Distribution Theory. Lecture Notes in Mathematics, vol. 1239. Springer, Berlin (1987)

    Book  Google Scholar 

  30. Vojta, P.: A refinement of Schmidt’s subspace theorem. Am. J. Math. 111, 489–518 (1989)

  31. Vojta, P.: On Cartan’s theorem and Cartan’s conjecture. Am. J. Math. 119, 1–17 (1997)

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Acknowledgements

We would like to thank the referee for carefully reading our manuscript and for his/her valuable comments on the first version of this paper, which help us to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Hanoi National University of Education, 136-Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Duc Quang Si

  2. Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam

    Duc Quang Si

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  1. Duc Quang Si
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Correspondence to Duc Quang Si.

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Dedicated to Professor Le Mau Hai on the occasion of His 70th birthday.

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Si, D.Q. Quantitative subspace theorem and general form of second main theorem for higher degree polynomials. manuscripta math. 169, 519–547 (2022). https://doi.org/10.1007/s00229-021-01329-z

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