On the diophantine equation \(f(a^m,y)=b^n\).

*(English)*Zbl 0963.11020Let \(f(X,Y)\) 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.

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.

Reviewer: Günter Lettl (Graz)

##### MSC:

11D61 | Exponential Diophantine equations |