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Polynomial-exponential equations involving several linear recurrences. (English) Zbl 1064.11031

Let \(A\) be a multiplicative semigroup of complex numbers and let \(\mathcal E_A\) denote the ring of complex sequences of the form \[ G_n = c_1\alpha_1^n + c_2\alpha_2^n +\cdots +c_t\alpha_t^n , \] where the \(\alpha\)’s belong to \(A\) and \(c_1\), \(c_2\), …, \(c_t\) are complex numbers. The authors consider polynomial-exponential equations of the form \[ G_n^{(0)}\,y^d + G_n^{(1)}\,y^{d-1}+ \cdots + G_n^{(d-1)}\,y +G_n^{(d)}=0. \] They show that, under certain (natural) conditions, such an equation has only finitely many solutions. This generalizes results of Corvaja and Zannier on similar equations.
The two main tools in their proof are the subspace theorem of Schmidt-Schlickewei and a suitable version of the implicit function theorem for analytic functions of several variables.
This main result has several corollaries, we choose one which is easy to state:
Corollary. Let \(a\) and \(b\) be non-zero algebraic numbers and \(\alpha\), \(\beta\) be positive integers. Then the equation \[ y^3+a\alpha^n +b \beta^n =0 \] has only finitely many solutions if \(\alpha^3 \neq \beta^2\); but if \(\alpha^3 = \beta^2\) then the set of integer solutions \((n,y)\) is contained in the set \[ \{(n,c\beta^{n/3}) : n \in \mathbb N,\; c^3+ac+b=0\}. \]

MSC:

11D61 Exponential Diophantine equations
11D45 Counting solutions of Diophantine equations
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