Number of solutions of the equation \(| x^ d- c^ z y^ d| =p\)
Let \(\alpha_ 1,\dots, \alpha_ k\) be pairwise non-equivalent primitive elements of a number field \(K\) of degree \(d\). Let \(P_ 1, \dots, P_ k\in \mathbb{Z} [x,y]\) be homogeneous irreducible polynomials with \(P(\alpha_ i, 1)=0\) for \(i=1,\dots, k\). The author proves, that if \(d\geq 4\), then for almost all \(h\in \mathbb{N}\) the sum of the number of solutions of the equations \[ | P_ i (x,y) |=h \qquad \text{in} \quad x,y\in \mathbb{Z} \] is \(\leq 2d^{t(h)}\), where \(t(h)\) is the number of distinct prime factors of \(h\). Except for finitely many right hand sides, this theorem improves the result of \textit{E. Bombieri} and \textit{W. M. Schmidt} [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)] for \(d\geq 4\). As a corollary it is shown, that for fixed \(c\in \mathbb{N}\) and prime \(d>3\), the equation \[ | x^ d- c^ z y^ d |=p \qquad \text{in} \quad x,y,z\in \mathbb{N} \] has at most \(d+1\) solutions with \(c\nmid y\). The proofs involve some earlier results of the author [Comp. Math. 94, No. 1, 29-49 (1994; Zbl 0820.11020)], as well as the theorem of Evertse- Laurent-van der Poorten-Schlickewei on the solutions of \(S\)-unit equations.
scientific article; zbMATH DE number 703891
scientific article