论文标题
Markoff注射率的$ Q $ -ANALOG猜想
A $q$-analog of the Markoff injectivity conjecture holds
论文作者
论文摘要
Markoff三元组的元素是由某些由Christoffel单词定义的矩阵产品中的系数给出的，而Markoff Injodentive猜想是一个长期存在的开放问题（也称为独特性猜想），然后等同于对基督徒词语的注射性。最近提出了这些矩阵产品的$ q $ - analog，我们证明，基督徒词的注射率适用于这种$ q $ -Analog。证明基于$ q = \ exp（iπ/3）$的评估。 Unity的其他根源提供了一些有关原始问题的信息，这与情况相对应$ Q = 1 $。我们还将问题扩展到任意单词，并提供了一对不可能的单词。
The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a long-standing open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A $q$-analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this $q$-analog. The proof is based on the evaluation at $q = \exp(iπ/3)$. Other roots of unity provide some information on the original problem, which corresponds to the case $q=1$. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.
