We call weak Markoff theory the theory of Markoff restricted to integral quadratic forms (instead of real ones), and to quadratic real numbers (instead of general real numbers). We show that weak Markoff theory may be reduced to combinatorial properties of Christoffel words, avoiding the use of bi-infinite sequences. These properties are two new characterizations of Christoffel words, one using the lexicographical order, the other being of arithmetic nature, based on values of continuant polynomials.

中文翻译：

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Christoffel的话和弱的Markoff理论

我们称弱Markoff理论为Markoff的理论，它限于整数二次形式（而不是实数形式）和二次实数（而不是一般实数）。我们证明了弱的Markoff理论可以简化为Christoffel单词的组合特性，从而避免使用双无限序列。这些属性是Christoffel单词的两个新特征，一个是按字典顺序使用的，另一个是基于连续多项式值的算术性质。