The Markoff equation over polynomial rings

Abstract

When $A=3$, the positive integral solutions of the so-called Markoff equation

$$\begin{aligned} M_A:x^2 + y^2 + z^2 = Axyz \end{aligned}$$

can be generated from the single solution (1, 1, 1) by the action of certain automorphisms of the hypersurface. Since Markoff’s proof of this fact, several authors have showed that the structure of $M_A(R)$, when R is ${\mathbb Z}[i]$ or certain orders in number fields, behave in a similar fashion. Moreover, for $R={\mathbb Z}$ and $R={\mathbb Z}[i]$, Zagier and Silverman, respectively, have found asymptotic formulae for the number of integral points of bounded height. In this paper, we investigate these problems when R is a polynomial ring over a field K of odd characteristic. We characterize the set $M_A(K[t])$ in a similar fashion as Markoff and previous authors. We also give an asymptotic formula that is similar to Zagier’s and Silverman’s formula.