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Bivariate polynomial injections and elliptic curves

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Abstract

For every number field k, we construct an affine algebraic surface X over k with a Zariski dense set of k-rational points, and a regular function f on X inducing an injective map \(X(k)\rightarrow k\) on k-rational points. In fact, given any elliptic curve E of positive rank over k, we can take \(X=V\times V\) with V a suitable affine open set of E. The method of proof combines value distribution theory for complex holomorphic maps with results of Faltings on rational points in sub-varieties of abelian varieties.

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References

  1. An, T., Wang, J., Wong, P.-M.: Strong uniqueness polynomials: the complex case. Complex Var. Theory Appl. 49(1), 25–54 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Avanzi, R., Zannier, U.: The equation \(f(X)=f(Y)\) in rational functions \(X=X(t)\), \(Y=Y(t)\). Compositio Math. 139(3), 263–295 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Boneh, D., Corrigan-Gibbs, H.: Bivariate polynomials modulo composites and their applications. Advances in cryptology-ASIACRYPT 2014. Part I, 42-62, Lecture Notes in Computer Science, 8873, Springer, Heidelberg (2014)

  4. Browkin, J., Brzeziński, J.: Some remarks on the \(abc\)-conjecture. Math. Comput. 62(206), 931–939 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre. Math. Annalen 46, 481–512 (1895)

    MATH  Google Scholar 

  6. Carney, A., Hortsch, R., Zieve, M.: Near-injectivity of polynomial functions on number fields. In progress (2019)

  7. Cornelissen, G.: Stockage diophantien et hypothèse abc généralisée. C. R. Acad. Sci. Paris Sér. I Math. 328(1), 3–8 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)

    MathSciNet  MATH  Google Scholar 

  9. Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math. 133(3), 549–576 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Faltings, G.: The general case of S. Lang’s conjecture. In: Barsotti Symposium in Algebraic Geometry, Perspectives in Mathematics, vol. 15, pp. 175–182 (1994)

  11. Fueter, R., Pólya, G.: Rationale Abzählung der Gitterpunkte. Vierteljahrsschrift der Naturf. Gesellschaft in Zürich, vol. 58, pp. 380–386 (1923)

  12. Fujimoto, H.: On uniqueness of meromorphic functions sharing finite sets. Am. J. Math. 122(6), 1175–1203 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Green, M.: Some Picard theorems for holomorphic maps to algebraic varieties. Am. J. Math. 97, 43–75 (1975)

    MathSciNet  MATH  Google Scholar 

  14. Hayman, W.: Waring’s Problem für analytische Funktionen. Bayer. Akad. Wiss. Math. Natur. Kl. Sitzungsber. 1985, 1–13 (1984)

    MathSciNet  MATH  Google Scholar 

  15. Hu, P.-C., Li, P., Yang, C.-C.: Unicity of meromorphic mappings. Advances in Complex Analysis and its Applications, 1. Kluwer Academic Publishers, Dordrecht, (2003). x+467 pp. ISBN: 1-4020-1219-5

  16. Kawamata, Y.: On Bloch’s conjecture. Invent. Math. 57, 97–100 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Lew, J., Rosenberg, A.: Polynomial indexing of integer lattice-points. I. General concepts and quadratic polynomials. J. Number Theory 10(2), 192–214 (1978)

    MathSciNet  MATH  Google Scholar 

  18. Lew, J., Rosenberg, A.: Polynomial indexing of integer lattice-points. II. Nonexistence results for higher-degree polynomials. J. Number Theory 10(2), 215–243 (1978)

    MathSciNet  MATH  Google Scholar 

  19. Poonen, B.: Multivariable polynomial injections on rational numbers. Acta Arithmetica 2(145), 123–127 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Rémond, G.: Décompte dans une conjecture de Lang. Invent. Math. 142(3), 513–545 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Rémond, G.: Sur les sous-variétés des tores. Compos. Math. 134(3), 337–366 (2002)

    MATH  Google Scholar 

  22. Wang, L.-P., Zhang, X.: Partial Bits Exposure Attacks on a New Commitment Scheme Based on the Zagier Polynomial. In: International Conference on Information Security and Cryptology, pp. 357–366. Springer, Cham, (2016)

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Acknowledgements

The results in this work were motivated by discussions at the PUC number theory seminar and I deeply thank the attendants. I also thank Gunther Cornelissen and Michael Zieve for comments on an earlier version of this work, and the anonymous referees for corrections and several valuable suggestions that improved the presentation.

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  1. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, 4860 Av. Vicuña Mackenna, Macul, RM, Chile

    Hector Pasten

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  1. Hector Pasten
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Correspondence to Hector Pasten.

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This research was supported by FONDECYT Regular Grant 1190442.

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Pasten, H. Bivariate polynomial injections and elliptic curves. Sel. Math. New Ser. 26, 22 (2020). https://doi.org/10.1007/s00029-020-0548-x

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