×

The Markoff equation \(X^ 2+Y^ 2+Z^ 2=aXYZ\) over quadratic imaginary fields. (English) Zbl 0702.11012

It is known that all solutions of the Markov-type equation \[ (*)\quad X^ 2+Y^ 2+Z^ 2=aXYZ \] (0\(\neq a\in {\mathbb{Z}})\) in X,Y,Z\(\in {\mathbb{Z}}\) can be generated from an initial solution by means of simple transformations. The purpose of the paper is to consider (*) in orders R of imaginary quadratic fields. The main theorem gives a similar description of all solutions X,Y,Z\(\in R\) of (*) for any \(a\in R\), \(| a| \geq 3\). It turns out, that solutions exist only if the imaginary quadratic field is \({\mathbb{Q}}(i)\). Moreover, in case \(R={\mathbb{Z}}[i]\), \(a\in {\mathbb{Z}}[i]\) with \(| a| \geq 4\), the author gives an asymptotic formula for the number of solutions X,Y,Z\(\in {\mathbb{Z}}[i]\) of (*) with height \(\leq H\). Such an asymptotic formula was proved in the classical case by D. Zagier [Math. Comput. 39, 709-723 (1982; Zbl 0501.10015)].
Reviewer: I.Gaál

MSC:

11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 0501.10015
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cassels, J. W.S., (An Introduction to Diophantine Approximation (1957), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0077.04801
[2] Cohn, H., Minimal geodesics on Fricke’s torus-covering, (Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 73-86 · Zbl 0455.30030
[3] Evertse, J.-H., On sums of \(S\)-units and linear recurrences, Compositio Math., 53, 225-244 (1984) · Zbl 0547.10008
[4] Markoff, A. A., Sur les formes binaires indéfinies, Math. Ann., 17, 379-399 (1880) · JFM 12.0143.02
[5] Zagier, D., On the number of Markoff numbers below a given bound, Math. Comp., 39, 709-723 (1982) · Zbl 0501.10015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.