多项式环上的Markoff方程
The Markoff equation over polynomial rings
摘要：
当$A=3$时，所谓的Markoff方程$$M_A:x^2 + y^2 + z^2 = Axyz$$的正整数解可以通过超曲面的某些自同构的作用从单解$(1,1,1)$生成.自从Markoff证明了这一事实，一些作者已经证明了$M_A(R)$的结构，当$R$是$\mathbf{Z}[i]$或数域中的某些顺序时，表现出类似的方式。此外，对于$R=\mathbf{Z}$和$R=\mathbf{Z}[i]$，Zagier和Silverman分别找到了有界高度的积分点数的渐近公式。本文研究了当$R$是奇特征域$K$上的多项式环时的这些问题。我们的特点集$M_A(K[t])$在类似的方式Markoff和以前的作者。我们还给出了一个渐近公式，类似于Zagier和Silverman的公式。
英文摘要：
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.
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