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A generalization of the subspace theorem for higher degree polynomials in subgeneral position

    In this paper, we prove a generalization of Schmidt’s subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety over a number field. Our result improves and generalizes the previous results on Schmidt’s subspace theorem for the case of higher degree polynomials.

    AMSC: 11J68, 11J25, 11J97


    • 1. H. Cartan, Sur les zéroes des combinaisons linéaries de p fonctions holomorphes données, Mathematica 7 (1933) 80–103. Google Scholar
    • 2. Z. Chen, M. Ru and Q. Yan, The degenerated second main theorem and Schmidts subspace theorem, Sci. China Math. 55 (2012) 1367–1380. CrossrefGoogle Scholar
    • 3. P. Corvaja and U. Zannier, On a general Thue’s equation, Amer. J. Math. 126 (2004) 1033–1055. Crossref, ISIGoogle Scholar
    • 4. J. Evertse and R. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Not. 25 (2002) 1295–1330. CrossrefGoogle Scholar
    • 5. J. Evertse and R. Ferretti, A Generalization of the Subspace Theorem with Polynomials of Higher Degree, Developments in Mathematics, Vol. 16 (Springer, New York, 2008), pp. 175–198. CrossrefGoogle Scholar
    • 6. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52 (Springer, New York, 1977). CrossrefGoogle Scholar
    • 7. A. Levin, On the Schmitd subspace theorem for algebraic points, Duke Math. J. 263 (2014) 2841–2885. CrossrefGoogle Scholar
    • 8. E. I. Nochka, On the theory of meromorphic functions, Sov. Math. Dokl. 27 (1983) 377–381. Google Scholar
    • 9. C. F. Osgood, A number theoretic-differential equations approach to generalizing Nevanlinna theory, Indian J. Math. 23 (1981) 1–15. Google Scholar
    • 10. C. F. Osgood, Sometimes effective Thue–Siegel–Roth–Schmidt–Nevanlinna bounds, or better, J. Number Theory 21 (1985) 347–389. Crossref, ISIGoogle Scholar
    • 11. S. D. Quang, Schmidts subspace theorem for moving hypersurfaces, Int. J. Number Theory 14 (2018) 103–121. LinkGoogle Scholar
    • 12. S. D. Quang, Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces, Trans. American Math. Soc. 371(4) (2019) 2431–2453; Crossref, ISIGoogle Scholar
    • 13. K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955) 1–20. CrossrefGoogle Scholar
    • 14. M. Ru and P. M. Wong, Integral points of n-{2n + 1 hyperplanes in general position}, Invent. Math. 106 (1991) 195–216. Crossref, ISIGoogle Scholar
    • 15. H. P. Schlickewei, The p-adic Thue–Siegel–Roth–Schmidt theorem, Arch. Math. 29 (1977) 267–270. CrossrefGoogle Scholar
    • 16. W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970) 189–201. Crossref, ISIGoogle Scholar
    • 17. W. M. Schmidt, Norm form equations, Ann. of Math. 96 (1972) 526–551. CrossrefGoogle Scholar
    • 18. W. M. Schmidt, Simultaneous approximation to algebraic numbers by elements of a number field, Monatsh. Math. 79 (1975) 55–66. CrossrefGoogle Scholar
    • 19. W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, Vol. 785 (Springer, New York, 1980). Google Scholar
    • 20. P. Vojta, Diophantine Approximation and Value Distribution Theory, Lecture Notes in Mathematics, Vol. 1239 (Springer, Berlin, 1987). CrossrefGoogle Scholar
    • 21. P. Vojta, A refinement of Schmidts subspace theorem, Amer. J. Math. 111 (1989) 489–518. Crossref, ISIGoogle Scholar
    • 22. E. A. Wirsing, On approximations of algebraic numbers by algebraic numbers of bounded degree, in 1969 Number Theory Institute, Proceedings of Symposia in Pure Mathematics, Vol. 20 (American Mathematical Society, Providence, RI, 1971), pp. 213–247. Google Scholar
    Published: 27 December 2018
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