沿连分数递减高度
Decreasing height along continued fractions
摘要：
欧几里德算法最终终止的事实在数学中是普遍存在的。在连分数的语言中，可以这样表述：高斯映射x-->{1/x}下的有理点的轨道最终达到零。在二次数域上定义的高斯映射的这一事实的类比与平移曲面上的流理论有关，并且已经通过强大的机器建立，最终依赖于维奇二分法。本文对二次不变迹域上的非余紧三角形群的每个可分解类，构造了一个Gauss映射，其定义矩阵生成该类中的一个群，并给出了一个直接且完备的终止性证明.作为一个副产品，我们提供了一个新的证明的事实，即非余紧三角形群的二次不变的迹域有该领域的投影线的尖点的交比集。我们的证明是基于分析的行动的非负矩阵的二次整数项目上的Weil高度的点。作为分析的结果，我们表明，我们的地图的字母表中的长符号序列可以有效地分裂成块的预定形状具有的属性，服从序列的点的高度，属于基本字段严格减少在每个块端。由于高度不能无限减小，因此终止属性随之而来。
英文摘要：
The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->{1/x} eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of noncocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that noncocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of nonnegative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.
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