The Markoff equation over polynomial rings

Abstract

When $A=3$, the positive integral solutions of the so-called Markoff equation $$M_A:x^2 + y^2 + z^2 = Axyz$$ 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 $\mathbf{Z}[i]$ or certain orders in number fields, behave in a similar fashion. Moreover, for $R=\mathbf{Z}$ and $R=\mathbf{Z}[i]$, 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.