When \(A=3\), the positive integral solutions of the so-called Markoff equation

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.

当\(A=3\) 时，所谓的马尔科夫方程的正积分解

可以通过超曲面的某些自同构的作用从单个解 (1, 1, 1) 生成。自从马尔科夫证明这一事实以来，几位作者已经表明\(M_A(R)\) 的结构，当R是\({\mathbb Z}[i]\)或数字字段中的某些顺序时，表现在类似的时尚。此外，对于\(R={\mathbb Z}\)和\(R={\mathbb Z}[i]\)，Zagier 和 Silverman 分别找到了有界高度积分点数的渐近公式。在本文中，我们研究了当R是奇特性域K上的多项式环时的这些问题。我们刻画了集合\(M_A(K[t])\)以与 Markoff 和以前的作者类似的方式。我们还给出了一个类似于 Zagier 和 Silverman 公式的渐近公式。