The Quotient Problem for Entire Functions

Guo, Ji

Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 479-489

Résumé

Let $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

Détail

DOI : 10.4153/S0008439518000097

Mots-clés : quotient problem, entire functions, second main theorem, linear recurrence

Bibliographie

[1] Borel, E., Sur les zéros des fonctions entières . Acta math. 20(1897), no. 1, 357–396. . Google Scholar | DOI

[2] Corvaja, P. and Zannier, U., Finiteness of integral values for the ratio of two linear recurrences . Invent. math. 149(2002), no. 2, 431–451. . Google Scholar | DOI

[3] Green, M., Some Picard theorems for holomorphic maps to algebraic varieties . Amer. J. Math. 97(1975), no. 1, 43–75. . Google Scholar | DOI

[4] Guo, J. and Wang, J.T.-Y., Asymptotic gcd and divisible sequences for entire functions . Trans. Amer. Math. Soc., to appear. Google Scholar

[5] Lang, S., Introduction to complex hyperbolic spaces . Springer-Verlag, New York, 1987. . Google Scholar | DOI

[6] Pasten, H. and Wang, J.T.-Y., GCD Bounds for analytic functions . Int. Math. Res. Not. IMRN (2017), no. 1, 47–95. . Google Scholar | DOI

[7] Van Der Poorten, A. J., Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles . C. R. Acad. Sci. Paris 302(1988), 97–102. Google Scholar

[8] Ru, M., Nevanlinna theory and its relation to Diophantine approximation . World Scientific, Publishing Co., Inc., River Edge, NJ, 2001. . Google Scholar | DOI

[9] Rumely, R., Notes on van der Poorten’s proof of the Hadamard quotient theorem . In: Séminaire de Théorie des Nombres, Paris 1986–87 , Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988, pp. 349–409. Google Scholar

[10] Vojta, P., On Cartan’s theorem and Cartan’s conjecture . Amer. J. Math. 119(1997), no. 1, 1–17. Google Scholar

[11] Vojta, P., Diophantine approximation and Nevanlinna theory . In: Arithmetic geometry , Lecture Notes in Mathematics, 2009, Springer-Verlag, Berlin, 2011, pp. 111–224. . Google Scholar | DOI

[12] Zannier, U., Diophantine equations with linear recurrences. An overview of some recent progress . J. Théor. Nombres Bordeaux 17(2005), 423–435. Google Scholar