Combinatorics of Words and Number Theory

单词组合学和数论

• 批准号: RGPIN-2020-04685
• 负责人: Reutenauer, Christophe
• 金额: $ 1.75万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717255
• 关键词: Combinatorics; Words; Number; Theory

One of the classical problems of mathematics is to find solutions to Diophantine Equations (named after Diophantus of Alexandria, a Greek mathematician of the third century AD). More precisely, only integral solutions are considered in this context (unlike e.g. x^2=2, whose solution is the square root of 2, a non-integer). One famous such equation is the equation x^2+y^2=z^2, which recalls the Pythagorean theorem, and whose infinitely many integral solutions (e.g. x=3,y=4,z=5) are all known. But an even more famous example is the equation x^n+y^n=z^n, appearing in the so-called "Fermat's last theorem". The fact that this equation has no solutions when the exponent n is at least equal to 3, is a result proved by Andrew Wiles in the 1990's. My program of research is based on a new original approach to the Markoff Diophantine equation x^2+y^2+z^2=3xyz. Markoff was a brilliant Russian number-theorist who studied this equation (when he was 20), around 1880. He became even more famous later, through the theory of Markov processes. He himself wrote his name with two 'f', in his first, French-written, articles. It turns out that one may completely solve the Markoff equation, using combinatorics on words, in a manner that sheds new light on many aspects of the question. Combinatorics on words is a rather new branch of mathematics, with links to automata theory, algebra and number theory, and also linguistics. The link with the Markoff equation exploits a certain construction on words, called palindromisation: by means of a certain algorithm, devised by Aldo de Luca, each word is turned into a palindrome. This construction, after some matrix computations, gives all triples which are solutions of the Markoff equation. I recently obtained this new solution of the equation in collaboration with two student of mine (Abram, Lapointe). My research program will explore many aspects of new links between combinatorics of words and number theory.

丢番图方程（以公元三世纪希腊数学家亚历山大的丢番图命名）的解是数学中的一个经典问题。更准确地说，在这种情况下只考虑积分解（不像x^2=2，其解是2的平方根，一个非整数）。其中一个著名的方程是方程x^2+y^2=z^2，它让人想起毕达哥拉斯定理，它的无穷多个积分解（例如x=3,y=4,z=5）都是已知的。但一个更著名的例子是方程x^n+y^n=z^n，出现在所谓的“费马大定理”中。当n的指数至少等于3时，这个方程没有解，这是安德鲁·怀尔斯在20世纪90年代证明的结果。