Z上仿射群的基本几何

摘要：

本文的主要内容是整数上的仿射群的几何，$\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ .构造了角、线段、三角形和椭圆的图灵可计算完全$\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$轨道不变量。在有理仿射$\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$ -几何中，椭圆通过Clifford-Hasse-Witt不变量，通过Hasse-Minkowski定理进行分类。我们分类椭圆在$\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ -几何结合的结果Apollonius的Perga和Pappus的亚历山德里亚与Hirzebruch-Jung连分数算法和Morelli-Wessodarczyk解决方案的弱织田猜想的因式分解的环面品种。然后我们考虑{ \it有理多面体}，即，$\mathbb R^n$中具有有理顶点的单形的有限并集。组合流形的马尔可夫不可识别性定理说明了两个有理多面体$P$和$P'$是否连续$\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$ -等可剖分的问题的不可判定性。对于$P$和$P'$的连续$\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$等剖分性，也存在同样的问题。证明了$\mathbb R^n$中两个有理多面体$P,Q$是否具有相同的$\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ -轨道的判定性.

英文摘要：

The subject matter of this paper is the geometry of the affine group over the integers, $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$. Turing-computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit invariants are constructed for angles, segments, triangles and ellipses. In rational affine $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-geometry, ellipses are classified by the Clifford--Hasse--Witt invariant, via the Hasse-Minkowski theorem. We classify ellipses in $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch-Jung continued fraction algorithm and the Morelli-Włodarczyk solution of the weak Oda conjecture on the factorization of toric varieties. We then consider {\it rational polyhedra}, i.e., finite unions of simplexes in $\mathbb R^n$ with rational vertices. Markov's unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra $P$ and $P'$ are continuously $\mathsf{GL}(n,\mathbb Q)\ltimes \mathbb Q^n$-equidissectable. The same problem for the continuous $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-equi\-dis\-sect\-ability of $P$ and $P'$ is open. We prove the decidability of the problem whether two rational polyhedra $P,Q$ in $\mathbb R^n$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$-orbit.

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