论文标题
多项式环上的Markoff方程
The Markoff equation over polynomial rings
论文作者
论文摘要
当$ a = 3 $时，所谓的Markoff方程的积极积分解决方案$$ m_a：x^2 + y^2 + z^2 = axyz $$ 可以通过超出表面的某些自动形态的作用从单个解决方案$（1,1,1）$生成。自从Markoff证明了这一事实的证明以来，几位作者表明，$ M_A（R）$的结构是$ R $是$ \ Mathbf {Z} [I] $或数字字段中某些订单的结构，以类似的方式行事。此外，对于$ r = \ mathbf {z} $和$ r = \ mathbf {z} [i] $，Zagier和Silverman，已经找到了有界高度积分点的渐近公式。在本文中，当$ r $是一个奇数特征的字段$ k $上的多项式戒指时，我们研究了这些问题。我们以与Markoff和以前的作者相似的方式来表征集合$ m_a（k [t]）$。我们还提供了类似于Zagier和Silverman的公式的渐近公式。
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.
