Markoff-Rosenberger triples in geometric progression (Q2439823)

Let \(K\) be an algebraic number field with ring of integers \(O_K\). The author studies solutions \(x,y,z\in O_K\) of the Markoff-Rosenberger equation \(ax^2+by^2+cz^2=dxyz\) where \(a,b,c,d\in O_K\), such that \(x,y,z\) form a geometric progression. Beside certain algorithmic and computational results, it is proved that infinitely many integral Markoff-Rosenberger triples in geometric progression over a number field \(K\) can exist only if either \(K=\mathbb{Q}\), or \(K\) is an imaginary quadratic field. When \(K=\mathbb{Q}\), an explicit description of the (finite) solution set is given. One of the main tools in the proofs is the computation of integer points of genus zero curves.