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On the diophantine equation $$f(a^m,y)=b^n$$. (English) Zbl 0963.11020
Let $$f(X,Y)$$ be a polynomial with rational coefficients of the form $f(X,Y) = a_0 Y^d + a_1(X) Y^{d-1} + \dots a_d(X)$ with $$0 \neq a_0 \in \mathbb Q$$, $$d \geq 2$$ and such that $$f(0,Y)$$ has no multiple roots. For natural numbers $$a,b$$, which are not relatively prime, the authors investigate the solutions $$(m,n,y) \in \mathbb Z^3$$ of the exponential Diophantine equation $f(a^m,y)=b^n \;. \tag{1}$ Their main result is: if (1) has infinitely many solutions with $$\min \{|m|, |n|\} \to \infty$$ then there exist a rational polynomial $$p(X)$$ and a non-zero integer $$h$$ such that $$f(X^h,p(X))$$ is a monomial (and, furthermore, $$a$$ and $$b$$ are multiplicatively dependent).
For the proof, three cases are distinguished depending on whether there exists a sequence of solutions with $$m/n$$ tending to a non positive limit, to a positive limit, or $$+ \infty$$, respectively. The first case is easy to deal with, for the second case one completes at a prime $$p \mid \gcd (a,b)$$ and considers $$Y$$, defined by $$f(x,Y)=z$$, as a $$p$$-adic function in the two variables $$x$$ and $$z$$, and in the third case one considers $$Y$$, defined by $$f(x,Y)=0$$, as an algebraic function over $$\mathbb Q(x)$$ and completes at infinity. Then W. M. Schmidt’s Subspace Theorem (one-dimensional, or $$p$$-adic, or projective, resp.) is used, and finally a Lemma is applied, which is based on a result of P. Liardet [Astérisque 24/25, 187-210 (1975; Zbl 0315.14005)].
The authors also show that all possible polynomials $$p(X)$$ and integers $$h$$, which appear in the main result, can be found effectively.

MSC:
 11D61 Exponential Diophantine equations
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