$\mathrm{GL}_2(\mathbb{Z})$ 上的费马型矩阵方程

中文摘要 AI 辅助

我们确定了对于每个整数 $n\ge3$，$\mathrm{GL}_2(\mathbb{Z})$ 中费马型矩阵方程 $X^n+Y^n=Z^n$ 的完整有序解集，其表述以同时整共轭轨道的形式给出，同时保持方程中变量的对称性。该方程可解当且仅当 $4\nmid n$ 且 $6\nmid n$。因此，与行列式为1的情形相反，$3$ 的奇数倍在完整群 $\mathrm{GL}_2(\mathbb{Z})$ 中是可解的，并且必然涉及行列式为 $-1$ 的分量。对于每个允许的偶指数，完整有序解集具有一个典范的不相交轨道分解，其索引与矩阵毕达哥拉斯方程中出现的同一横截集相同。对于满足 $3\nmid n$ 的奇指数，交换解恰好构成六个同时共轭轨道。当 $3\nmid n$ 时，非交换奇解恰好构成二十个同时共轭轨道；当 $3\mid n$ 时，恰好构成十四个。在由带符号变量置换和同时整共轭生成的更粗等价关系下，这些非交换轨道分别收缩为四类和三类。证明结合了矩阵毕达哥拉斯方程的典范轨道分类、Cayley--Hamilton 递推、交换子整除性、迹增长估计、挠矩阵的整共轭性，以及针对三次方程的一个初等对称约化迹论证。

英文摘要

We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $4\nmid n$ and $6\nmid n$. Thus, in contrast with the determinant-one setting, odd multiples of $3$ are solvable in the full group $\mathrm{GL}_2(\mathbb{Z})$ and necessarily involve determinant-$-1$ components. For every admissible even exponent, the full ordered solution set has a canonical disjoint orbit decomposition indexed by the same transversal that occurs in the matrix Pythagorean equation. For odd exponents with $3\nmid n$, the commuting solutions form exactly six simultaneous-conjugacy orbits. The noncommuting odd solutions form exactly twenty simultaneous-conjugacy orbits when $3\nmid n$, and exactly fourteen when $3\mid n$. Under the coarser equivalence generated by signed variable permutations and simultaneous integral conjugation, these noncommuting orbits collapse to four and three classes, respectively. The proof combines the canonical orbit classification of the matrix Pythagorean equation, Cayley--Hamilton recurrences, commutator divisibility, trace-growth estimates, integral conjugacy of torsion matrices, and an elementary symmetry-reduced trace argument for the cubic equation.

发表机构

• School of Mathematics and Statistics, Guangdong University of Foreign Studies（广东外语外贸大学数学与统计学院）
• School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）

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