Basic geometry of the affine group over Z

TL;DR

This paper explores the geometry of the affine group over integers, constructing computable invariants for geometric objects, classifying ellipses, and analyzing the decidability of polyhedra dissection problems within this group.

Contribution

It introduces computable invariants for affine group orbits, classifies ellipses using advanced algebraic methods, and establishes decidability results for rational polyhedra in integer affine geometry.

Findings

Constructed Turing-computable orbit invariants for various geometric figures.

Classified ellipses in integer affine geometry using algebraic invariants.

Proved the decidability of orbit equivalence for rational polyhedra under integer affine transformations.

Abstract

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-W\l odarczyk 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.

Equations

$$\gamma \in \mathsf{GL}(n,\mathbb{Z})⋉{\mathbb{Z}}^n$$

$$\widetilde{x}=\bigl{(}\operatorname{\mathrm{den}}(x)\cdot x_{1},\ldots,\,\operatorname{\mathrm{den}}(x)\cdot x_{n},\,\,\operatorname{\mathrm{den}}(x)\bigr{)}\in\mathbb{Z}^{n+1}$$

$$\mathrm{pos}[v_1,\dots ,v_s]=\{x\in {\mathbb{R}}^n\mid x={\rho }_0{v}_0+\cdots +{\rho }_s{v}_s,0\le {\rho }_0,\dots ,{\rho }_s\in \mathbb{R}\}.$$

$$\mathcal{P}(p_1,\dots ,p_m)=\{x\in {\mathbb{R}}^n\mid x={\rho }_1{p}_1+\cdots +{\rho }_m{p}_m,0\le {\rho }_1,\dots ,{\rho }_m<1\}$$

$$d_F=\min \{\mathrm{den}(v)\mid v\in F\cap {\mathbb{Q}}^n\}.$$

$$\mathcal{R}=\mathrm{pos}[{\tilde{v}}_0,{\tilde{r}}_1,...,{\tilde{r}}_e].$$

$$\mathrm{den}(w_j)>\mathrm{den}(w_{j1})>\mathrm{den}(w_{j2})>\cdots \ge \mathrm{den}(v_0),{\tilde{w}}_{jt}={\tilde{w}}_{jt-1}-{\tilde{v}}_0.$$

$$\mathrm{den}(w_{js})=\mathrm{den}(v_0)=d_F,$$

$$\mathcal{P}({\tilde{v}}_0,\dots ,{\tilde{v}}_e,\tilde{x})=\{{\mathbb{R}}^n\ni x={\rho }_0{\tilde{v}}_0+\cdots +{\rho }_e{\tilde{v}}_e+{\rho }_{e+1}\tilde{x}\mid 0\le {\rho }_0,...,{\rho }_{e+1}<1\}$$

$$\mathrm{conv}(v_0,\dots ,v_e,s)$$

$$\mathrm{den}(s)=c_F.$$

$$1\leq\tilde{u}_{n+1}=\operatorname{\mathrm{den}}(u)=\pm\tilde{s}_{n+1}+kd_{F}=\pm\operatorname{\mathrm{den}}(s)+kd_{F},\mbox{ for some integer $k$.}$$

$$\alpha \mapsto \mathcal{R}(\alpha )$$

$$(I,J{)}^0,(I,J{)}^1,\dots$$

$$\{e_0=\tilde{a},e_1,e_2,\dots ,e_u,e_{u+1}=\tilde{b}\}\subseteq {\mathbb{Z}}^{n+1}\subseteq {\mathbb{R}}^{n+1}$$

$${\tilde{x}}_i=e_i,(i=1,\dots ,u),x_0=a,x_{u+1}=b.$$

$$\mathsf{HJ}(A)=\mbox{ the triangulation of $\operatorname{\mathrm{conv}}(a,b)$ with vertices $x_{0},x_{1},\dots,x_{u},x_{u+1}$}$$

$$\gamma :\mathrm{conv}(a,b)\cong \mathrm{conv}(a^{\prime },b^{\prime }).$$

$${\lambda }_1(\mathrm{conv}(a,b))=\sum _{i=0}^u\frac{1}{\mathrm{den}(x_i)\mathrm{den}(x_{i+1})},$$

$${\Delta }_0=\Delta ,{\Delta }_1,\dots ,{\Delta }_{r-1},{\Delta }_r=\nabla$$

$${\lambda }_1(\mathrm{conv}(a,b))<{\lambda }_1(\mathrm{conv}(a,c),{\Delta }^{\prime }\cup {\Delta }^{\prime \prime })={\lambda }_1(\mathrm{conv}(a,c)).$$

$$\mathsf{side}(A)=\{c_{\mathrm{aff}(A)},{\lambda }_1(A),\mathrm{den}(a),\mathrm{den}(x_1)\}$$

$$\mathsf{hj}(S)=(\mathrm{den}(\mathsf{HJ}(S)),c_{\mathrm{aff}(S)}),$$

$${\gamma }_0={\gamma }_1=\cdots ={\gamma }_u.$$

$$l=m\mbox{ and for some $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$, }\gamma(x_{i})=x_{i}^{\prime}\mbox{ for all }i=1,2,\dots,l+1.$$

$${\lambda }_1(\mathrm{conv}(a^{\prime },\gamma (b)))<{\lambda }_1(\mathrm{conv}(a^{\prime },b^{\prime })).$$

$${\lambda }_1(\mathrm{conv}(a^{\prime },b^{\prime }))={\lambda }_1(\mathrm{conv}(a,b))={\lambda }_1(\mathrm{conv}(\gamma (a),\gamma (b)))={\lambda }_1(\mathrm{conv}(a^{\prime },\gamma (b))),$$

$$\gamma :\mathrm{conv}(u,v,w)\cong \mathrm{conv}(u^{\prime },v^{\prime },w^{\prime }).$$

$$\mathsf{tri}(T)=(\mathsf{side}(\mathrm{conv}(v,u)),\mathsf{angle}(H_{vu},K_{vw}),\mathsf{side}(\mathrm{conv}(v,w))).$$

$$(x_0=v,x_1,\dots ,x_l,x_{l+1}=u)\text{mbox}and(x_0^{\prime }=v^{\prime },x_1^{\prime },\dots ,x_m^{\prime },x_{m+1}^{\prime }=u^{\prime })$$

Full text

2000 Mathematics Subject Classification. Primary: 13A50. Secondary: 11D09, 11E16, 11H55, 13A50, 14G05, 14H50, 14L24, 14M25, 14R20, 20H25

1. Introduction

In Klein’s 1872 inaugural lecture at the University of Erlangen one finds the following programmatic statement111See page 219 in his paper “A comparative review of recent researches in geometry”, Bull. New York Math. Soc., 2(10) (1893) 215–249, https://projecteuclid.org/euclid.bams/1183407629:

As a generalization of geometry arises then the following comprehensive problem $[\dots]$: Given a manifoldness and a group of transformations of the same; to develop the theory of invariants relating to that group.

In the spirit of Klein’s program, in the first part of this paper we construct Turing-computable complete invariants of angles, segments, triangles and ellipses in the geometry of the affine group over the integers, $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$.

Our starting point is the following problem, for arbitrary $X,X^{\prime}\subseteq\mathbb{R}^{n}$:

[TABLE]

Otherwise stated: Do $X$ and $X^{\prime}$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit ? By a “decision method” for this problem we understand a Turing machine $\mathcal{M}$ which, over any input $X,X^{\prime}$ decides whether $X$ and $X^{\prime}$ have the same orbit. $X$ and $X^{\prime}$ must be effectively presented to $\mathcal{M}$ as finite strings of symbols. Thus, e.g., if $X$ is a triangle, we will assume that its vertices have rational coordinates, and $X$ is presented to $\mathcal{M}$ via the list of its vertices. If $X$ is an ellipse, $X$ is understood as the zeroset $Z(\phi)$ in $\mathbb{R}^{2}$ of a quadratic polynomial $\phi(x,y)$ with rational coefficients, and is presented to $\mathcal{M}$ via the list of coefficients of $\phi.$

In Theorem 5.4 we equip rational triangles with a (Turing-) computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit invariant. Basic constituents of our side-angle-side invariant are the invariants introduced in [4] for affine spaces in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry, (Theorem 2.5). Section 3 is devoted to showing the computability of these invariants. Further main constituents are the $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit invariants for angles and segments constructed in Theorems 4.2 and 5.3. It follows that Problem (1) is decidable for segments, angles and triangles in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry.

In Section 6 we construct computable complete invariants for ellipses: In euclidean geometry, ellipses are classified by the lengths of their major and minor axes. In $\mathsf{GL}(2,\mathbb{Q})\ltimes\mathbb{Q}^{2}$-geometry, the Hasse-Minkowski theorem classifies rational ellipses by their (Clifford-Hasse-Witt) invariants, [2, 1.1], [9, §5], [13, §4]. Let $\mathcal{E}$ denote the set of rational ellipses $E\subseteq\mathbb{R}^{2}$ containing a rational point. As is well known, (see, e.g., [6], [25]), from the input rational coefficients of $\phi$ it is decidable whether the zeroset of $\phi$ is an ellipse $E\in\mathcal{E}$. If this is the case, $E$ contains a dense set of rational points. In Theorem 6.4 a finite set of invariants is computed from $\phi$, in such a way that a rational ellipse $E^{\prime}$ has the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit of $E$ iff $E$ and $E^{\prime}$ have the same invariants. It follows that Problem (1) is decidable for ellipses.

For our constructions in this paper we combine results on conjugate diameters by Apollonius of Perga [1] and Pappus of Alexandria [21], with the Hirzebruch-Jung continued fraction algorithm, [7, 8, 20], and the Morelli-Włodarczyk solution of the weak Oda conjecture on the factorization of toric varieties. [15, 27],

In Theorem 7.2, Problem (1) is shown to be decidable for rational polyhedra, i.e., finite union of simplexes with rational vertices, [26]. In a final remark this result is compared with Markov’s theorem ([5, 24], see Theorem 7.1) on the unrecognizability of rational polyhedra in $\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$-geometry, to the effect that manifolds cannot be characterized up to homeomorphism by computable complete invariants.

2. Classification of rational affine spaces in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry

A rational affine hyperplane $H$ is a subset of $\mathbb{R}^{n}$ of the form H=\{z\in\mathbb{R}^{n}\mid\langle p,z\rangle=\upsilon\},\mbox{ for some nonzero vector p\in\mathbb{Q}^{n}$and$\upsilon\in\mathbb{Q}.} Here $\langle\mbox{-},\mbox{-}\rangle$ denotes scalar product. Any intersection of rational affine hyperplanes in $\mathbb{R}^{n}$ is said to be a rational affine space in $\mathbb{R}^{n}$. For any subset $X$ of $\mathbb{R}^{n}$, the affine span $\,\,\operatorname{\mathrm{aff}}(X)\,\,$ is defined by stipulating that a point $z$ belongs to $\operatorname{\mathrm{aff}}(X)$ iff there are $w_{1},\ldots,w_{k}\in X$ and $\lambda_{1},\ldots,\lambda_{k}\in\mathbb{R}$ such that $\lambda_{1}+\dots+\lambda_{k}=1$ and $z=\lambda_{1}w_{1}+\dots+\lambda_{k}w_{k}$. (See [11] for this terminology.) Equivalently, we say that $\operatorname{\mathrm{aff}}(X)$ is the set of affine combinations of elements of $X$. A set $\{y_{1},\ldots,y_{l}\}\subseteq\mathbb{R}^{n}$ is said to be affinely independent if none of its elements is an affine combination of the remaining elements. For $\,\,0\leq d\leq n$, a d-simplex in $\mathbb{R}^{n}$ is the convex hull $T=\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{d})$ of $d+1$ affinely independent points $v_{0},\ldots,v_{d}\in\mathbb{R}^{n}$. The vertices $v_{0},\ldots,v_{d}$ are uniquely determined by $T$. $T$ is said to be a rational simplex if its vertices are rational. The (affine) dimension $\dim(T)$ is equal to $d$.

The denominator   $\operatorname{\mathrm{den}}(x)$   of a rational point $x=(x_{1},\dots,x_{n})\in\mathbb{Q}^{n}$ is the least common denominator of its coordinates. The vector

[TABLE]

is said to be the homogeneous correspondent of $x$. The integer vector $\widetilde{x}$ is primitive, [20, p.24], i.e., the greatest common divisor of its coordinates is equal to 1. Every primitive integer vector $q\in\mathbb{Z}^{n+1}$ whose $(n+1)$th coordinate is $>0$ is the homogeneous correspondent of a unique rational point $x\in\mathbb{R}^{n}$, called the affine correspondent of $q.$

With the notation of [7, I, Definition 1.9, p.6], given vectors $v_{1},\ldots,v_{s}\in\mathbb{R}^{n}$ we write

[TABLE]

for their positive hull in $\mathbb{R}^{n}$. Let $t=1,2,\ldots,n$. Adopting the terminology of [7, p.146], by a $t$-dimensional rational simplicial cone in $\mathbb{R}^{n}$ we understand a set $C\subseteq\mathbb{R}^{n}$ of the form $C=\operatorname{\mathrm{pos}}[w_{1},\ldots,w_{t}],$ for linearly independent primitive integer vectors $w_{1},\ldots,w_{t}\in\mathbb{Z}^{n}$. The latter are said to be the primitive generating vectors of $C$. They are uniquely determined by $C$. By a *face * of $C$ we mean the positive hull of a subset of $\{w_{1},\ldots,w_{t}\}.$ The face of $C$ determined by the empty set is the singleton $\{0\}$. This is the only zero-dimensional cone in $\mathbb{R}^{n}$.

Farey regularity

A rational $d$-simplex $T=\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{d})\subseteq\mathbb{R}^{n}$ is said to be (Farey)* regular* (“unimodular” in [17]) if the set $\{\tilde{v}_{0},\ldots,\tilde{v}_{d}\}$ of homogeneous correspondents of its vertices can be extended to a basis of the free abelian group $\mathbb{Z}^{n+1}$. Equivalently, the cone ${\operatorname{\mathrm{pos}}}[\tilde{v}_{0},\dots,\tilde{v}_{m}]\subseteq\mathbb{R}^{n+1}$ is regular in the sense of [7, Definition V 1.10, p.146], or “nonsingular” in the sense of [8, p.29] and [20, p.15], or “unimodular” in the sense of [11, §7].

A rational triangulation in $\mathbb{R}^{n}$ is an (always finite) simplicial complex $\Delta$ such that the vertices of every simplex in $\Delta$ have rational coordinates in $\mathbb{R}^{n}$. The point-set union of all simplexes in $\Delta$ (called the support of $\Delta$) is the most general possible rational polyhedron in $\mathbb{R}^{n}$, [26, Chapter II].

A simplicial complex is said to be a regular triangulation (of its support) if all its simplexes are regular. Regular triangulations are affine counterparts of regular fans of toric algebraic geometry, [7, p.165], called “nonsingular fans” in [20, Theorem 1.10]. (Warning: the notion of a “regular” triangulation given in [11, p. 387] has a different meaning.)

Regular triangulations play a fundamental role in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry, as well as in the theory of AF C*-algebras with lattice-ordered $K_{0}$-group, [16, 18], (also see [19] for recent developments).

Lemma 2.1** (A corollary of Steinitz exchange lemma).**

Let $\operatorname{\mathrm{conv}}(x_{0},\dots,x_{m})\subseteq\mathbb{R}^{n}$ be a regular $m$-simplex. Suppose $\operatorname{\mathrm{conv}}(y_{0},\dots,y_{e})\subseteq\mathbb{R}^{n}$ is a regular $e$-simplex and $\operatorname{\mathrm{aff}}(y_{0},\dots,y_{e})=\operatorname{\mathrm{aff}}(x_{0},\dots,x_{e})$. Then $e\leq m$, and  $\operatorname{\mathrm{conv}}(y_{0},\dots,y_{e},x_{e+1},\dots,x_{m})$ is a regular $m$-simplex.

Proof.

Passing to the homogeneous correspondents $\tilde{x}_{j},\tilde{y}_{k}\in\mathbb{Z}^{n+1}$ of these rational points, by definition of regularity we have a trivial variant of Steinitz exchange lemma. ∎

Lemma 2.2** (A corollary of Minkowski convex body theorem).**

Given linearly independent integer vectors $p_{1},\dots,p_{m}\in\mathbb{R}^{n}$ let

[TABLE]

be their half-open parallelepiped. Then $\{p_{1},\dots,p_{m}\}$ is part of a basis of the free abelian group $\mathbb{Z}^{n}$ iff the only integer point in $\mathcal{P}(p_{1},\dots,p_{m})$ is the origin $0\in\mathbb{R}^{n}$.

Proof.

[3, VII, Corollary 2.6]. ∎

$\mathcal{P}(p_{1},\dots,p_{m})$ is called the fundamental parallelepiped of $p_{1},\dots,p_{m}$ in [11, §7].

The following result is a routine consequence of Lemma 2.2. We include the proof for later use in Lemma 4.1(iii).

Lemma 2.3**.**

Suppose $p\in\mathbb{Z}^{m+1}$, and $\{b_{1},\dots,b_{m},q\}$ is a basis of the free abelian group $\mathbb{Z}^{m+1}$. Then $\{b_{1},\dots,b_{m},p\}$ is a basis of $\mathbb{Z}^{m+1}$ iff $p=\pm q+l$ for some linear combination $l$ of $b_{1},\dots,b_{m}$ with integer coefficients.

Proof.

$(\Rightarrow)$ Let us denote by $[b_{1},\dots,b_{m},p]$ the unimodular matrix with columns vectors $b_{1},\dots,b_{m},\,p$. Let $H=\mathbb{R}b_{1}+\dots+\mathbb{R}b_{m}$ be the linear span of the vectors $b_{1},\dots,b_{m}$ in $\mathbb{R}^{m+1}.$ Let $H^{\pm}$ be the affine hyperplanes in $\mathbb{R}^{m+1}$ parallel to $H$ and containing $\pm q$ respectively. By way of contradiction, suppose $p$ does not coincide with $\pm q+l$ for any a linear combination $l$ of $b_{1},\dots,b_{m}$ with integer coefficients. Since $|\det[b_{1},\dots,b_{m},p]|=1=$ volume of $\mathcal{P}(b_{1},\dots,b_{m},p)=$ volume of $\mathcal{P}(b_{1},\dots,b_{m},q)$, then $p\in H^{+}$ or $p\in H^{-}$, say $p\in H^{+}.$ Then for some linear combination $l^{\prime}$ of $b_{1},\dots,b_{m}$ with integer coefficients, the translated half-open parallelepiped $q+l^{\prime}+\mathcal{P}(b_{1},\dots,b_{m})$ contains the integer point $p\not=q+l^{\prime}.$ So $\mathcal{P}(b_{1},\dots,b_{m})$ contains the nonzero integer point $p-q-l^{\prime}.$ By Lemma 2.2, this contradicts the fact that $\{b_{1},\dots,b_{m}\}$ is part of a basis of $\mathbb{Z}^{m+1}$. The $(\Leftarrow)$ direction is trivial. ∎

The invariants $d_{F}$ and $c_{F}$

For every rational affine space $F\subseteq\mathbb{R}^{n}$ we set

[TABLE]

Next suppose $F$ is $e$-dimensional. If $0\leq e<n$ we define the integer $c_{F}>0$ as the least possible denominator $\operatorname{\mathrm{den}}(v)$ of a rational point $v\in\mathbb{Q}^{n}$ such that there are points $v_{0},\ldots,v_{e}\in F\cap\mathbb{Q}^{n}$ making $\operatorname{\mathrm{conv}}(v,v_{0},\ldots,v_{e})$ a regular $(e+1)$-simplex. If $e=n$ we define $c_{F}=1$.

Lemma 2.4**.**

Let $F\subseteq\mathbb{R}^{n}$ be a rational affine space. If $\dim(F)\not=n-1$, $c_{F}=1$. If $\dim(F)=n-1$, then $1\leq c_{F}\leq\max(1,d_{F}/2)$ and $\gcd(c_{F},d_{F})=1$, where “gcd” denotes greatest common divisor.

Proof.

[4, Lemma 9]. ∎

The following theorem states that the triplet $(\dim(F),d_{F},c_{F})$ is a complete invariant for the rational affine space $F$ in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry:

Theorem 2.5** (Rational affine spaces in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry).**

Let   $F$  and $G$  be rational affine spaces in $\mathbb{R}^{n}$. Then   $F$  and   $G$   have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit iff $(\dim(F),d_{F},c_{F})=(\dim(G),d_{G},c_{G})$.

Proof.

[4, Theorem 8]. ∎

3. The computation of $(\dim(F),d_{F},c_{F})$ for

$F$ an affine rational space

Lemma 3.1**.**

There exists a Turing machine $\mathcal{T}$ with the following property: For any two $(n+1)$-tuples $V=(v_{0},\ldots,v_{n})$ and $W=(w_{0},\ldots,w_{n})$ of rational points in $\mathbb{R}^{n}$ with $\operatorname{\mathrm{den}}(v_{i})=\operatorname{\mathrm{den}}(w_{i}),\,\,(i=0,\ldots,n)$, $\mathcal{T}$ decides whether both $\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{n})$ and $\operatorname{\mathrm{conv}}(w_{0},\ldots,w_{n})$ are regular $n$-simplexes in $\mathbb{R}^{n}$ and, if this is the case, computes the uniquely determined map $\gamma=\phi_{VW}\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\gamma(v_{i})=w_{i}$ for each $i\in\{0,\ldots,n\}$.

Proof.

Lemma 2.2 yields a decision procedure to check whether $\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{n})$ is regular. If this is the case, the proof of [4, Lemma 1] yields an effective procedure to compute the desired map $\gamma$. ∎

Lemma 3.2**.**

There is a Turing machine which, given a rational affine space $F=\operatorname{\mathrm{aff}}(a_{1},\dots,a_{m})\subseteq\mathbb{R}^{n},\,\,(m\mbox{ arbitrary, each }a_{i}\in\mathbb{Q}^{n})$ together with a rational point $v_{0}\in F$ with $\operatorname{\mathrm{den}}(v_{0})=d_{F}\,,$ first computes the integer $e=\dim(F)$ and then outputs points $v_{1}\ldots,v_{e}\in\mathbb{Q}^{n}\cap F$ with $\operatorname{\mathrm{den}}(v_{1})=\dots=\operatorname{\mathrm{den}}(v_{e})=d_{F}$ such that $\operatorname{\mathrm{conv}}(v_{0},v_{1},\dots,v_{e})$ is a regular $e$-simplex.

Proof.

The dimension $e$ of $F$ is immediately computed from the input rational points $a_{i}$. We can easily pick rational points $r_{1},\dots,r_{e}\in F$ such that the set $R=\operatorname{\mathrm{conv}}(v_{0},r_{1},\dots,r_{e})$ is an $e$-simplex. With the notation of (2), let the set $\mathcal{R}\subseteq\mathbb{R}^{n+1}$ be defined by

[TABLE]

The desingularization procedure [7, VI, 8.5], [8, p.48] yields a complex $\Phi$ of rational polyhedral cones (for short, a fan) in $\mathbb{R}^{n+1}$ such that each $(e+1)$-dimensional cone $C\in\Phi$ has the form $C={\rm pos}[\tilde{q}_{0},\tilde{q}_{1},\dots,\tilde{q}_{e}]$ for a suitable set $\{\tilde{q}_{0},\tilde{q}_{1},\dots,\tilde{q}_{e}\}$ of primitive integer vectors which is part of a basis of the free abelian group $\mathbb{Z}^{n+1}.$ $\Phi$ is known as a regular (or nonsingular) fan providing a desingularization of $\mathcal{R}.$ $\Phi$ is computable by a Turing machine over input $v_{0},r_{1},\dots,r_{e}$. The rational points $q_{0},q_{1},\dots,q_{e}\in F$ are the vertices of a regular complex $\Delta$ with support $R$. Thus in particular $\Delta$ contains a regular $e$-simplex $T_{0}=\operatorname{\mathrm{conv}}(v_{0},w_{1},\ldots,w_{e})$ having $v_{0}$ among its vertices. By construction, the set $\{\widetilde{v}_{0},\widetilde{w}_{1},\ldots,\widetilde{w}_{e}\}$ is part of a basis of the free abelian group $\mathbb{Z}^{n+1}.$

If $\operatorname{\mathrm{den}}(w_{i})=d_{F}$ for all $i=1,\dots,e$ we are done. Otherwise, let $j$ be the smallest index in $\{1,\dots,e\}$ such that $\operatorname{\mathrm{den}}(w_{j})>\operatorname{\mathrm{den}}(v_{0})$. Then the integer vector $\widetilde{w}_{j}-\widetilde{v}_{0}$ is primitive, because replacing $\widetilde{w}_{j}$ by $\widetilde{w}_{j}-\widetilde{v}_{0}$ in the set $\{\widetilde{v}_{0},\widetilde{w}_{1},\ldots,\widetilde{w}_{e}\}$ we obtain a part of a basis of $\mathbb{Z}^{n+1}$. So let the rational point $w_{j1}$ be defined by $\widetilde{w}_{j1}=\widetilde{w}_{j}-\widetilde{v}_{0}.$ Since both $w_{j}$ and $v_{0}$ lie in $F$, then so does $w_{j1}.$ Further, the $e$-simplex $T_{j1}=\operatorname{\mathrm{conv}}(v_{0},w_{2},\dots,w_{j1},\dots,w_{e})\subseteq F$ is regular, and $\operatorname{\mathrm{den}}(w_{j})>\operatorname{\mathrm{den}}(w_{j1})\geq\operatorname{\mathrm{den}}(v_{0}).$ Inductively, we have regular $e$-simplexes $T_{j1},T_{j2},\ldots\subseteq F$ with

[TABLE]

After a finite number $s=s_{j}\geq 0$ of steps we will have $\operatorname{\mathrm{den}}(w_{js})\leq\operatorname{\mathrm{den}}(v_{0})$, whence

[TABLE]

by the assumed minimality property of $\operatorname{\mathrm{den}}(v_{0})$. We then set $v_{j}=w_{js}\,\,\,\mbox{ and }\,\,T_{1}=T_{js}\,,$ and note that $T_{1}\subseteq F$ is the regular $e$-simplex obtained from $T_{0}$ replacing $w_{j}$ by the new vertex $v_{j}\in F$. Assuming inductively that $T_{r+1}$ is obtained in a similar way by replacing a vertex of $T_{r}$ by a new vertex lying in $F$ with denominator equal to $d_{F}$, the procedure will finally output the desired regular $e$-simplex $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})\subseteq F$ with $\operatorname{\mathrm{den}}(v_{0})=\dots=\operatorname{\mathrm{den}}(v_{e})=d_{F}\,.$ The computability of the map $(a_{1},\dots,a_{j})\mapsto(v_{0},\dots,v_{e})$ is clear. ∎

Theorem 3.3**.**

Let $F\subseteq\mathbb{R}^{n}$ be an $e$-dimensional rational affine space $(e=0,\dots,n)$. Then there are rational points $v_{0},\ldots,v_{n}\in F$ such that

• (i)

$\operatorname{\mathrm{den}}(v_{i})=d_{F}$*   for all   $i\in\{0,\dots,e\}$;*

• (ii)

$\operatorname{\mathrm{den}}(v_{i})=c_{F}$*   for all   $i\in\{e+1,\dots,n\}$;*

• (iii)

$\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{n})$* is a regular $n$-simplex. *

Moreover, once $F$ is presented as   $\operatorname{\mathrm{aff}}(a_{0},\dots,a_{e})$ for some $a_{0},\dots,a_{e}\in\mathbb{Q}^{n}$, the points $v_{0},\dots,v_{e}$ can be computed by a Turing machine.

Proof.

The problem whether $F$ contains rational points of a prescribed denominator is decidable, and whenever a solution exists, such a point can be explicitly found—e.g., via integer linear programming, [11, §7]. Thus we first check whether $F$ contains some integer point. If such $x$ exists then $d_{F}$=1. Otherwise we proceed inductively to check if $F$ contains a point with denominator 2, 3, $\dots$. Since $F$ is a rational subspace of $\mathbb{R}^{n}$, this process terminates, yielding a point $v_{0}\in F$ with the smallest possible denominator. Thus $d_{F}=\operatorname{\mathrm{den}}(v_{0})$. Now Lemma 3.2 yields a regular $e$-simplex $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})\subseteq F$ satisfying $\operatorname{\mathrm{den}}(v_{1})=\dots=\operatorname{\mathrm{den}}(v_{e})=\operatorname{\mathrm{den}}(v_{0})=d_{F}.$ The proof now proceeds arguing by cases:

Case 1: $F$ has codimension 1, i.e., $e=n-1$.

Let $C_{0}$ be a closed cube with rational vertices, centered at $v_{0}$ and containing the simplex $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})$. Let $C_{0}\subseteq C_{1}\subseteq C_{2}\subseteq\dots$ be a sequence of closed $n$-cubes with rational vertices, centered at $v_{0}$, where $C_{t+1}$ is obtained by doubling the sides of $C_{t}$. For any $t=0,1,\dots,$ we check whether there exists a rational point $s\in C_{t}$ satisfying the conditions

1. (*)

$\operatorname{\mathrm{den}}(s)\leq\max(1,d_{F}/2)$, and 2. (**)

the set $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e},s)$ is a regular $n$-simplex.

Each cube $C_{t}$ contains only finitely many rational points $x$ satisfying $\operatorname{\mathrm{den}}(x)\leq\max(1,d_{F}/2).$ For any such point $x$, Lemma 2.2 yields a method to decide whether $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e},x)$ is a regular $n$-simplex: one checks that the half-open $(n+1)$-dimensional parallelepiped

[TABLE]

does not contain any nonzero integer point. Lemma 2.4 in combination with [4, Lemma 7] ensures the existence of a point $s\in\mathbb{R}^{n}$ with $\operatorname{\mathrm{den}}(s)=c_{F}\leq\max(1,d_{F}/2)$, together with points $v^{*}_{0},\dots,v^{*}_{e}\in F$, all with denominator $d_{F}$, such that $\operatorname{\mathrm{conv}}(v^{*}_{0},\dots,v^{*}_{e},s)$ is a regular $n$-simplex. Similarly, since $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})\subseteq F$ is regular, an application of Lemma 2.1 shows that

[TABLE]

We have just shown that there is $t=1,2,\dots,$ and a rational point $s\in C_{t}$ satisfying conditions (*) and (**) above. This result is now strengthened as follows:

[TABLE]

If $d_{F}=1$ then $c_{F}=1$ and by Condition (*) we are done.

So assume $d_{F}\geq 2.$

If $n=1$ then $e=0$, so $F=\{v_{0}\}$ for some $v_{0}\in\mathbb{Q}\setminus\mathbb{Z}$ $d_{F}=\operatorname{\mathrm{den}}(v_{0})\geq 2$. Whenever $s\in\mathbb{Q}$ is such that $\operatorname{\mathrm{conv}}(v_{0},s)$ is a regular 1-simplex in $\mathbb{R}$ and $\operatorname{\mathrm{den}}(s)\leq\max(1,\operatorname{\mathrm{den}}(v_{0})/2)=\operatorname{\mathrm{den}}(v_{0})/2$, there is no $r$ with $\operatorname{\mathrm{conv}}(v_{0},r)$ regular and $\operatorname{\mathrm{den}}(r)<\operatorname{\mathrm{den}}(s)$. As a matter of fact, say without loss of generality $s>v_{0}.$ Repeated applications of Lemma 2.2 show: If $v_{0}<r<s$ then $\mathcal{P}(\tilde{v}_{0},\tilde{s})$ contains the integer point $\tilde{r}$, against the regularity of $\operatorname{\mathrm{conv}}(v_{0},s)$. If $v_{0}<s<r$ then $\mathcal{P}(\tilde{v}_{0},\tilde{r})$ contains the integer point $\tilde{s}$, against the regularity of $\operatorname{\mathrm{conv}}(v_{0},r)$. If $v_{0}>r$ then $\mathcal{P}(\tilde{v}_{0},\tilde{s})$ contains the integer point $\tilde{v}_{0}-\tilde{r}$, against the regularity of $\operatorname{\mathrm{conv}}(v_{0},r)$. This settles the case $n=1.$

If $n>1$ we can write $n-1=e\geq 1\mbox{ and }\operatorname{\mathrm{den}}(s)\leq d_{F}/2.$ Let \mathcal{D}=\{u\in\mathbb{Q}^{n}\mid\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e},u)\mbox{ is a regular n-simplex}\}. Thus $u\in\mathcal{D}$ iff $\{\tilde{v}_{0},\dots,\tilde{v}_{e},\tilde{u}\}$ is a basis of $\mathbb{Z}^{n+1}$. By Condition (**) and Lemma 2.3, $u\in\mathcal{D}$ iff $\tilde{u}=\pm\tilde{s}+c,$ for some linear combination $c$ of $\tilde{v}_{0},\dots,\tilde{v}_{e}$ with integer coefficients. Thus in particular, if $u\in\mathcal{D}$ the $(n+1)$th coordinate $\tilde{u}_{n+1}$ of $\tilde{u}$ satisfies

[TABLE]

This is so because the $(n+1)$th coordinates of $\tilde{v}_{0},\dots,\tilde{v}_{e}$ are all equal to $d_{F}.$ Since $\operatorname{\mathrm{den}}(s)\leq d_{F}/2$ then $\operatorname{\mathrm{den}}(u)\geq\operatorname{\mathrm{den}}(s)$ for all $u\in\mathcal{D}.$ Since, by (5), ${s^{*}}\in\mathcal{D}$ then $c_{F}\leq\operatorname{\mathrm{den}}(s)\leq\operatorname{\mathrm{den}}(s^{*})=c_{F}$, which settles (6), and concludes the proof of Case 1.

*Case 2: * The codimension of $F$ is different from 1.

Then by Lemma 2.4, $c_{F}=1$. Using Lemma 3.2 we compute a regular simplex $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})\subseteq F$ with $\operatorname{\mathrm{den}}(v_{1})=\dots=\operatorname{\mathrm{den}}(v_{e})=\operatorname{\mathrm{den}}(v_{0})=d_{F}.$ In case $\dim(\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e}))=n$ we are done. In case $\dim(\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e}))\not=n$, knowledge that $c_{F}=1$ simplifies the search (within the increasing sequence of cubes $C_{t}$) of the desired integer points $v_{e+1},\dots,v_{n}$ such that $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{n})$ is regular. Regularity amounts to the unimodularity of the integer matrix whose rows are the vectors $v_{0},\dots,v_{n}$—a decidable problem. Since each $C_{t}$ contains only finitely many integer points, an exhaustive search in each $n$-cube $C_{t}$ centered at $v_{0}$ will provide the desired points $v_{e+1},\dots,v_{n}$.

By construction, the map $(a_{0},\dots,a_{e})\mapsto(v_{0},\dots,v_{n})$ is computable. ∎

Corollary 3.4**.**

For any rational affine space $F\subseteq\mathbb{R}^{n}$ the invariants $\dim(F),\,\,d_{F}$ and $c_{F}$ in Theorem 2.5 are computable. Thus it is decidable whether the affine spans of two sets of points $a_{0},\ldots,a_{m}\in\mathbb{Q}^{n}$ and   $b_{0},\ldots,b_{l}\in\mathbb{Q}^{n}$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit. Moreover, there is a Turing machine $\mathcal{M}$ which, whenever two $e$-dimensional rational affine spaces $F,F^{\prime}\subseteq\mathbb{R}^{n}$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit ($e=0,\dots,n$), computes two $(n+1)$-tuples $V=(v_{0},\dots,v_{n})$ and $V^{\prime}=(v^{\prime}_{0},\dots,v^{\prime}_{n})$ of rational points with $\operatorname{\mathrm{den}}(v_{0})=\dots=\operatorname{\mathrm{den}}(v_{e})=\operatorname{\mathrm{den}}(v^{\prime}_{0})=\dots=\operatorname{\mathrm{den}}(v^{\prime}_{e})=d_{F}$ and $\operatorname{\mathrm{den}}(v_{e+1})=\dots=\operatorname{\mathrm{den}}(v_{n})=\operatorname{\mathrm{den}}(v^{\prime}_{e+1})=\dots=\operatorname{\mathrm{den}}(v^{\prime}_{n})=c_{F}$ such that $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{n})$ and $\operatorname{\mathrm{conv}}(v^{\prime}_{0},\dots,v^{\prime}_{n})$ are regular $n$-simplexes and the map $\gamma=\phi_{VV^{\prime}}\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of Lemma 3.1 sends $F$ onto $F^{\prime}$.

Proof.

From Theorem 3.3 we obtain: (i) rational points $v_{0},\dots,v_{e}\in F$ such that $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{e})$ is a regular $e$-simplex and $\operatorname{\mathrm{den}}(v_{0})=\dots=\operatorname{\mathrm{den}}(v_{e})=d_{F}$; (ii) (if $e<n$) additional points $v_{e+1},\dots,v_{n}\in\mathbb{R}^{n}$ such that $\operatorname{\mathrm{conv}}(v_{0},\dots,v_{n})$ is a regular $n$-simplex with $\operatorname{\mathrm{den}}(v_{e+1})=\dots=\operatorname{\mathrm{den}}(v_{n})=c_{F}$. Similarly, from $(b_{0},\ldots,b_{k})$, we get a regular $n$-simplex $\operatorname{\mathrm{conv}}(v^{\prime}_{0},\dots,v^{\prime}_{n})$ with $\operatorname{\mathrm{conv}}(v^{\prime}_{0},\dots,v^{\prime}_{e})\subseteq F^{\prime}$, and denominators as in (i)-(ii). By Theorem 2.5, the identity $(\dim(F),d_{F},c_{F})=(\dim(F^{\prime}),d_{F^{\prime}},c_{G^{\prime}})$ can be effectively checked. If the identity holds, Lemma 3.1 and Theorem 3.3 yield the desired Turing machine $\mathcal{M}$. ∎

Remark 3.5**.**

One might speculate that the map $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $F$ onto $F^{\prime}$ in Corollary 3.4 is obtainable by solving a system of equations $p_{1}=0,\dots,p_{k}=0,$ where each $p_{i}$ is a polynomial with integer coefficients and the unknowns are integers representing the terms the matrix $\gamma.$ As $n$ grows, so does the degree of the system. We are then faced with a formidable subproblem of a diophantine problem whose general undecidability was proved by Matiyasevič in his negative solution of Hilbert Tenth Problem, [14]. Taking an alternative route, the decidability of the orbit problem for $F$ and $F^{\prime}$ has been established by constructing suitable regular simplexes in $F$ and $F^{\prime}$, using the classification Theorem 2.5. In the next sections, refinements of these techniques will provide computable complete invariants for triangles and ellipses in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry.

4. Classification of

angles in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry

As a special case of a rational affine space, a rational line $L$ in $\mathbb{R}^{n}$ is a line containing at least two distinct rational points. Every rational point $v\in L$ determines two rational half-lines in $L$ with a common origin $v$. A rational oriented angle is a pair $(H,K)$ of rational half-lines in $\mathbb{R}^{n}$ with a common origin. We will henceforth assume that ($n\geq 2$ and) the affine spans of $H$ and $K$ in $\mathbb{R}^{n}$ are distinct (for short, the angle $(H,K)$ is nontrivial). Nontriviality is decidable by elementary linear algebra. We denote by $\widehat{HK}$ the convex portion of the plane $\operatorname{\mathrm{aff}}(H\cup K)\subseteq\mathbb{R}^{n}$ obtained by rotating $H$ to $K$ around $v$ in $\operatorname{\mathrm{aff}}(H\cup K)$, with the orientation from $H$ to $K$. Given a rational oriented angle $(H^{\prime},K^{\prime})$ in $\mathbb{R}^{n}$ we write $(H,K)\cong(H^{\prime},K^{\prime})$ if there is $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\gamma(H)=H^{\prime}$ and $\gamma(K)=K^{\prime}.$ When this is the case we also write $\gamma\colon(H,K)\cong(H^{\prime},K^{\prime})$. While $\widehat{HK}=\widehat{KH}$, Theorem 4.2 will show that the condition $(H,K)\cong(K,H)$ generally fails.

Lemma 4.1**.**

For any rational half-line $H\subseteq\mathbb{R}^{n}$ with origin $v$, let $H_{\rm reg}$ be the set of rational points $h\in H$ such that the segment $\operatorname{\mathrm{conv}}(v,h)$ is regular. We then have:

(i) Any two distinct points $h,k\in H_{\rm reg}$ have different denominators.

(ii) $H_{\rm reg}$ contains a farthest point from $v$, denoted $\mathsf{q}_{H}.$ This is also characterized as the point in $H_{\rm reg}$ with the smallest possible denominator.

(iii) Let $(H,K)$ be a rational oriented angle in $\mathbb{R}^{n}$ (with vertex $v$). Let $\mathcal{L}_{HK}$ be the set of rational points $y\in\widehat{HK}$ such that $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},y)$ is regular and $\operatorname{\mathrm{den}}(y)$ is as small as possible. Then there exists a (necessarily unique) point $\mathsf{p}_{HK}\in\mathcal{L}_{HK}$ nearest to $K$.

Proof.

It is easy to see that $H_{\rm reg}$ is an infinite set of rational points having $v$ as an accumulation point.

(i) By way of contradiction, assume $h$ and $k$ are distinct points of $H_{\rm reg}$ with $\operatorname{\mathrm{den}}(h)=\operatorname{\mathrm{den}}(k)$. Passing to homogeneous correspondents in $\mathbb{R}^{n+1}$ and recalling (3), it follows that either parallelogram $\mathcal{P}(\tilde{v},\tilde{h})$ or $\mathcal{P}(\tilde{v},\tilde{k})$, say $\mathcal{P}(\tilde{v},\tilde{h})$, contains a nonzero integer point. By Lemma 2.2, $\{\tilde{v},\tilde{h}\}$ cannot be extended to a basis of $\mathbb{Z}^{n+1}$, i.e., the segment $\operatorname{\mathrm{conv}}(v,h)$ is not regular—a contradiction.

(ii) immediately follows from (i) and Lemma 2.2.

(iii) Since any two points in $\mathcal{L}_{HK}$ have equal denominators, the (infinite) set $\mathcal{L}_{HK}$ has no accumulation points. From the proof of Lemma 2.3 it follows that $\mathcal{L}_{HK}$ is contained in a uniquely determined rational half-line $M\subseteq\widehat{HK}$ parallel to $H$, whose origin lies in ${K}$. This ensures the existence and uniqueness of the point $\mathsf{p}_{HK}$ nearest to $K$. ∎

The following theorem provides a computable complete invariant for rational oriented angles $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry:

Theorem 4.2** (Rational oriented angles in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry).**

For any rational oriented angle $(H,K)$ in $\mathbb{R}^{n}$ with vertex $v$, let   $\mathsf{angle}(H,K)$ be the following sextuple:

• (i)

The triple of integers $(\operatorname{\mathrm{den}}(v),\operatorname{\mathrm{den}}(\mathsf{q}_{H})$, $\operatorname{\mathrm{den}}(\mathsf{p}_{HK}))$.

• (ii)

The (first two) barycentric coordinates of $\mathsf{q}_{K}$ with respect to the oriented triangle $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK})$.

• (iii)

The integer $c_{\operatorname{\mathrm{aff}}(\widehat{HK})},$ (which, by Lemma 2.4, is dispensable when $n\not=3$).

Then the map    $(H,K)\mapsto\mathsf{angle}(H,K)$    is computable. Given a rational oriented angle $(H^{\prime},K^{\prime})$ in $\mathbb{R}^{n}$, we have $(H,K)\cong(H^{\prime}K^{\prime})$    iff    $\mathsf{angle}(H,K)=\mathsf{angle}(H^{\prime},K^{\prime})$. Thus the orbit problem (1) for rational angles in $\mathbb{R}^{n}$ is decidable.

Proof.

The definition of the rational points $\mathsf{q}_{H},\mathsf{p}_{HK},\mathsf{q}_{K}$ in Lemma 4.1 ensures their computability. The barycentric coordinates of $\mathsf{q}_{K}$ with respect to $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK})$ are rational and computable. The computability of the sextuple $\mathsf{angle}(H,K)$ now follows by Corollary 3.4.

In order to prove completeness of the invariant, let us suppose $\eta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ maps $H$ onto $H^{\prime}$ and $K$ onto $K^{\prime}$, $\eta\colon(H,K)\cong(H^{\prime},K^{\prime}).$ It follows that $\eta$ maps $\widehat{HK}$ onto $\widehat{H^{\prime}K^{\prime}}.$ Since $\eta$ preserves regular simplexes and denominators of rational points, Lemma 4.1 yields the identities $\eta(\mathsf{q}_{H})=\mathsf{q}_{H^{\prime}}$, $\eta(\mathsf{q}_{K})=\mathsf{q}_{K^{\prime}}$ and $\eta(\mathsf{p}_{HK})=\mathsf{p}_{H^{\prime}K^{\prime}}$. The denominators of $v,\,\,\mathsf{q}_{H},\,\,\mathsf{p}_{HK},\,\,\mathsf{q}_{K}$ respectively coincide with the denominators of $v^{\prime},\,\,\mathsf{q}_{H^{\prime}},\,\,\mathsf{p}_{H^{\prime}K^{\prime}},\,\,\mathsf{q}_{K^{\prime}}$. Since $\eta$ preserves affine combinations, the points $\mathsf{q}_{K}$ and $\mathsf{q}_{K^{\prime}}$ have the same barycentric coordinates with respect to the triangles   $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK})$   and   $\operatorname{\mathrm{conv}}(v^{\prime},\mathsf{q}_{H^{\prime}},\mathsf{p}_{H^{\prime}K^{\prime}})$. Since $\eta$ maps $\operatorname{\mathrm{aff}}(\widehat{HK})$ onto $\operatorname{\mathrm{aff}}(\widehat{H^{\prime}K^{\prime}})$, Theorem 2.5 yields $c_{\operatorname{\mathrm{aff}}(\widehat{HK})}=c_{\operatorname{\mathrm{aff}}(\widehat{H^{\prime}K^{\prime}})}$. Thus $\mathsf{angle}(H,K)$   =  $\mathsf{angle}(H^{\prime},K^{\prime})$.

Conversely, assume $\mathsf{angle}(H,K)=\mathsf{angle}(H^{\prime},K^{\prime})$, with the intent of proving $(H,K)\cong(H^{\prime},K^{\prime}).$ From the pair $(H,K)$ we compute the subspace $F=\operatorname{\mathrm{aff}}(\widehat{HK})$. Since $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK})$ is a regular 2-simplex, combining Lemma 2.1 and Theorem 3.3, we obtain an $(n-2)$-tuple ${\bf a}=(a_{1},\dots,a_{n-2})$ of rational points in $\mathbb{R}^{n}$, all with denominator $c_{F}$, such that $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK},{\bf a})$ is a regular $n$-simplex in $\mathbb{R}^{n}$. Letting $F^{\prime}=\operatorname{\mathrm{aff}}(\widehat{H^{\prime}K^{\prime}})$, we similarly compute an $(n-2)$-tuple ${\bf a}^{\prime}=(a^{\prime}_{1},\dots,a^{\prime}_{n-2})$ of rational points in $\mathbb{R}^{n}$, all with denominator $c_{F^{\prime}}$, in such a way that $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H^{\prime}},\mathsf{p}_{H^{\prime}K^{\prime}},{\bf a}^{\prime})$ is a regular $n$-simplex in $\mathbb{R}^{n}$. Since $\mathsf{angle}(H,K)=\mathsf{angle}(H^{\prime},K^{\prime})$, the denominators of the vertices of $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK},{\bf a})$ are pairwise equal to the denominators of the vertices of $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H^{\prime}},\mathsf{p}_{H^{\prime}K^{\prime}},{\bf a}^{\prime})$. Lemma 3.1 now yields a uniquely determined map $\theta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\theta(\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK},{\bf a}))=\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H^{\prime}},\mathsf{p}_{H^{\prime}K^{\prime}},{\bf a}^{\prime}).$ In more detail, $\theta(\mathsf{q}_{H})=\mathsf{q}_{H^{\prime}}$ and $\theta(v)=v^{\prime}$, and hence $\theta(H)=H^{\prime}.$ (Should we choose others $(n-2)$-tuples ${\bf b},{\bf b}^{\prime}$ of points with denominator $c_{\operatorname{\mathrm{aff}}(\widehat{HK})}=c_{\operatorname{\mathrm{aff}}(\widehat{H^{\prime}K^{\prime}})}$, such that $\operatorname{\mathrm{conv}}(v,\mathsf{q}_{H},\mathsf{p}_{HK},{\bf b})$ and $\operatorname{\mathrm{conv}}(v^{\prime},\mathsf{q}_{H^{\prime}},\mathsf{p}_{H^{\prime}K^{\prime}},{\bf b}^{\prime})$ are regular $n$-simplexes, the resulting map $\theta^{\prime}\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ given by Lemma 3.1 would still agree with $\theta$ on $\operatorname{\mathrm{aff}}(\widehat{HK})$.) Since $\theta$ preserves affine combinations and $\theta(\mathsf{p}_{HK})=\mathsf{p}_{H^{\prime}K^{\prime}}$, from $\mathsf{angle}(H,K)=\mathsf{angle}(H^{\prime},K^{\prime})$ it follows that $\theta(\mathsf{q}_{K})=\mathsf{q}_{K^{\prime}}$. We have just proved $\theta\colon(H,K)\cong(H^{\prime},K^{\prime})$, as desired to complete the proof of the theorem. ∎

Remark 4.3**.**

In $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry vertical angles need not have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit. For instance, let $L$ be the $x$-axis in $\mathbb{R}^{2}$, and $M\subseteq\mathbb{R}^{2}$ be the line passing through the points $v=(3/5,0)$ and $w=(1,1)$. Let $L^{\prime}\subseteq L$ be the half-line originating at $v$ along the positive direction of $L$. Let the half-line $L^{\prime\prime}$ be defined by $L^{\prime\prime}=\operatorname{\mathrm{cl}}(L\setminus L^{\prime})$, where “cl” denotes closure. Let $M^{\prime}\subseteq M$ be the half-line originating at $v$ and lying in the first quadrant. Let $M^{\prime\prime}=\operatorname{\mathrm{cl}}(M\setminus M^{\prime}).$ A straightforward computation shows that $\mathsf{angle}(L^{\prime\prime},M^{\prime\prime})\not=\mathsf{angle}(L^{\prime},M^{\prime})\not=\mathsf{angle}(M^{\prime\prime},L^{\prime\prime})$. Therefore, the two vertical angles $\widehat{L^{\prime}M^{\prime}}$ and $\widehat{L^{\prime\prime}M^{\prime\prime}}$ do not have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit.

While prima facie our computable complete invariant $\mathsf{angle}$ in Theorem 4.2 may look less elementary than its euclidean counterpart,222if indeed arc length or the circular functions are more elementary than our orbit-invariant $\mathsf{angle}$ in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry. the following proposition shows that any other computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit invariant of rational oriented angles in $\mathbb{R}^{n}$ is Turing-equivalent to our invariant $\mathsf{angle}$.

Proposition 4.4** (Universal property of the invariant $\mathsf{angle}$).**

Suppose $\mathsf{newangle}$ is a computable complete invariant of rational oriented angles in $\mathbb{R}^{n}$. Then there is a Turing machine $\mathcal{R}$ which, over any input string $\alpha=\mathsf{newangle}(H,K)$ outputs the string $\mathcal{R}(\alpha)=\mathsf{angle}(H,K)$. Conversely there is a Turing machine $\mathcal{S}$ which, over any input string $\beta=\mathsf{angle}(M,N)$, outputs the string $\mathcal{S}(\beta)=\mathsf{newangle}(M,N)$. Further,

[TABLE]

Proof.

Suppose $\alpha=\mathsf{newangle}(H,K)$ for some rational angle $(H,K)$. Equipping with some lexicographic order the set of all strings denoting rational angles in $\mathbb{R}^{n}$ and letting

[TABLE]

be their enumeration in this order, after a finite number of steps the first oriented rational angle $(I,J)^{t}$ satisfying $\alpha=\mathsf{newangle}((I,J)^{t})$ will be detected. This follows from our assumption about $\alpha$ and the computability of $\mathsf{newangle}.$ The computability and completeness of both $\mathsf{angle}$ and $\mathsf{newangle}$ now yield a Turing machine $\mathcal{R}$ computing, over input $\alpha$, the transformation $\alpha\mapsto(I,J)^{t}\mapsto\mathsf{angle}((I,J)^{t})=\mathsf{angle}(H,K).$ (In case $\alpha\not=\mathsf{newangle}((I,J)^{k})$ for all rational angles $(I,J)^{k},$     $\mathcal{R}$ will enter an infinite loop. We are not assuming that the range of the invariant $\mathsf{newangle}$ is decidable. So in general we cannot upgrade $\mathcal{R}$ to a machine $\mathcal{R}^{+}$ that terminates after a finite number of steps over any possible input.)

Conversely, suppose $\beta=\mathsf{angle}(M,N)$. The computability of $\mathsf{angle}$ similarly yields an effective procedure to detect the first oriented rational angle $(I,J)^{r}$ in the list (8) such that $\beta=\mathsf{angle}((I,J)^{r})$. Again, the computability and completeness of both invariants $\mathsf{newangle}$ and $\mathsf{angle}$ yield a Turing machine $\mathcal{S}$ computing the transformation $\beta\mapsto(I,J)^{r}\mapsto\mathsf{newangle}((I,J)^{r})=\mathsf{newangle}(M,N).$ Finally, (7) follows from the completeness of the invariants $\mathsf{newangle}$ and $\mathsf{angle}.$ ∎

With the same proof, the computable complete invariant for segments, triangles and ellipses constructed in the next two sections have the same universal property.

5. Classification of triangles in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry

The Hirzebruch-Jung algorithm:

notation and terminology

For any pair $(a,b)$ of distinct points in $\mathbb{Q}^{n}$ let us equip the rational segment $A=\operatorname{\mathrm{conv}}(a,b)\subseteq\mathbb{R}^{n}$ with the orientation from $a$ to $b$. $A$ is said to be an oriented rational segment. With the notation of (2), let $C=\operatorname{\mathrm{pos}}[\tilde{a},\tilde{b}]\subseteq\mathbb{R}^{n+1}$ be the positive hull of the homogeneous correspondents of $a$ and $b$. Let further  $N$ be the set of nonzero integer points in $C$, and $\mathsf{H}=\operatorname{\mathrm{conv}}(N)$, with its relative boundary $\partial\mathsf{H}$. Following [20, p.24-25] or [8, 2.6] we write

[TABLE]

for the set of integer points lying on the compact edges of $\partial\mathsf{H}$, listed in the order $\sqsubseteq$ from $\tilde{a}$ to $\tilde{b}$. Thus $e_{i}\sqsubseteq e_{j}$ iff the angle $\widehat{e_{0}0e_{i}}$ is contained in the angle $\widehat{e_{0}0e_{j}}$. The complex of cones (i.e., the fan) in $\mathbb{R}^{n+1}$ whose primitive generating vectors are the integer vectors ${e}_{i}$ is said to be obtained via the Hirzebruch-Jung (continued fraction, desingularization) algorithm on the cone $C$, [8, p.46]. Let the points $x_{i}\in\mathbb{Q}^{n}$ be defined by

[TABLE]

We will use the notation

[TABLE]

listed in the order inherited from $\sqsubseteq$.

Proposition 5.1**.**

The Hirzebruch-Jung algorithm of the oriented rational segment $A=\operatorname{\mathrm{conv}}(a,b)\subseteq\mathbb{R}^{n}$ outputs a list of rational points $a=x_{0},x_{1},\dots,x_{u},x_{u+1}=b$ having the following properties, for each $i=0,\dots,u$:

• (i)

The segment $\operatorname{\mathrm{conv}}(x_{i},x_{i+1})$ is regular.

• (ii)

$x_{i+1}$* is the (necessarily unique) rational point $z\in A$ with the smallest possible denominator such that the segment $\operatorname{\mathrm{conv}}(x_{i},z)$ is regular. Equivalently, $x_{i+1}$ is the farthest point $z$ from $x_{i}$ in $A$ such that $\operatorname{\mathrm{conv}}(x_{i},z)$ is regular.*

• (iii)

The vertices of $\mathsf{HJ}(A)$ are a subset of the vertices of every regular triangulation of $A$. Thus the list $x_{0},x_{1},\dots,x_{u},x_{u+1}$ is uniquely determined by $(a,b).$

• (iv)

The map $A\mapsto\mathsf{HJ}(A)$ is computable.

Proof.

This is the affine counterpart of [8, 2.6] or [20, Proposition 1.19]. The computability of the map $A\mapsto\mathsf{HJ}(A)$ is evident by definition of the Hirzebruch-Jung algorithm. ∎

The Hirzebruch-Jung regular triangulation of the segment $\operatorname{\mathrm{conv}}(-1/2,5/8)$ is shown in Figure 1.

Blow-up and blow-down

Suppose $\Delta$ and $\nabla$ are two simplicial complexes in $\mathbb{R}^{n}$ with the same support. We say that $\nabla$ is a subdivision of $\Delta$ if every simplex of $\nabla$ is contained in some simplex of $\Delta$. Let $\Delta$ be a simplicial complex and $c\in|\Delta|$. The blow-up $\Delta_{(c)}$ of $\Delta$ at $c$ is the simplicial complex in $\mathbb{R}^{n}$ obtained by the following procedure ([27, p. 376], [7, III, 2.1]):

Replace every simplex $S\in\Delta$ such that $c\in S$ by the set of all simplexes of the form $\operatorname{\mathrm{conv}}(c,F)$, where $F$ is any face of $S$ such that $c\notin F$.

Note that $\Delta_{(c)}$ is a subdivision of $\Delta$ with the same support of $\Delta$. The inverse of a blow-up is called a blow-down.

For any $m\geq 1$ and regular $m$-simplex $T=\operatorname{\mathrm{conv}}(v_{0},\ldots,v_{m})\subseteq\mathbb{R}^{n}$, the Farey mediant of $T$ is the affine correspondent of the vector $\tilde{v}_{0}+\cdots+\tilde{v}_{m}\in\mathbb{Z}^{n+1}$, where each $\tilde{v}_{i}$ is the homogeneous correspondent of ${v}_{i}$. In the particular case when $\Delta$ is a regular triangulation and $c$ is the Farey mediant of a simplex of $\Delta$, the blow-up $\Delta_{(c)}$ is regular.

The $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-invariant measure $\lambda_{1}$

The second main tool for the classification of rational segments in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry is the one-dimensional fragment $\lambda_{1}$ of the rational measure $\lambda_{d}$ introduced in [17, Theorem 2.1].

For any oriented rational segments $\operatorname{\mathrm{conv}}(a,b)$ and $\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ in $\mathbb{R}^{n}$ we write $\operatorname{\mathrm{conv}}(a,b)\cong conv(a^{\prime},b^{\prime})$ if there is $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\gamma(a)=a^{\prime}$ and $\gamma(b)=b^{\prime}$, in symbols,

[TABLE]

Theorem 5.2**.**

For any rational oriented segment $\operatorname{\mathrm{conv}}(a,b)\subseteq\mathbb{R}^{n}$ let

[TABLE]

where $a=x_{0},\,x_{1},\dots,x_{u},x_{u+1}=b$ are the consecutive vertices of the Hirzebruch-Jung desingularization $\mathsf{HJ}(\operatorname{\mathrm{conv}}(a,b))$. We then have:

• (i)

(Invariance)* $\operatorname{\mathrm{conv}}(a,b)\cong\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ $\Rightarrow$ $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b))=\lambda_{1}(\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime}))$.*

• (ii)

(Computability)* The map $(a,b)\mapsto\lambda_{1}(\operatorname{\mathrm{conv}}(a,b))$ is computable.*

• (iii)

(Independence)* Let $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\nabla)$ denote the result of the computation of $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b))$ in (10) by means of a regular triangulation $\nabla$ of $\operatorname{\mathrm{conv}}(a,b)$, in place of $\mathsf{HJ}(\operatorname{\mathrm{conv}}(a,b))$. Then $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b))$ = $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\nabla)$.*

• (iv)

(Monotonicity)* For every rational point $c\in\mathbb{R}^{n}$ with $\operatorname{\mathrm{conv}}(a,b)\subsetneqq\operatorname{\mathrm{conv}}(a,c)$ we have $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b))<\lambda_{1}(\operatorname{\mathrm{conv}}(a,c)).$*

Proof.

(i)-(ii) These are immediate consequences of Proposition 5.1(i)-(ii), because every map $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ preserves denominators and regularity.

(iii) Let $\Delta=\mathsf{HJ}(\operatorname{\mathrm{conv}}(a,b))$. Let $\Delta_{(e)}$ be obtained by blowing-up $\Delta$ at the Farey mediant $e$ of some $1$-simplex $S=\operatorname{\mathrm{conv}}(x_{i},x_{i+1})\in\Delta$. From the regularity of $S$ we get $\operatorname{\mathrm{den}}(e)=\operatorname{\mathrm{den}}(x_{i})+\operatorname{\mathrm{den}}(x_{i+1}).$ Then a routine verification shows that $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\Delta)=\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\Delta_{(e)}).$ The affine version of the Morelli-Włodarczyk theorem on decomposition of birational toric maps (solution of the weak Oda conjecture, [15], [27, 13.3]), yields a sequence

[TABLE]

of regular triangulations of $\operatorname{\mathrm{conv}}(a,b)$ such that for each $t=0,\ldots,r-1$, $\Delta_{t+1}$ is obtained from $\Delta_{t}$ by a blow-up at the Farey mediant of some simplex of $\Delta_{t}$, or vice versa, with the roles of $t$ and $t+1$ interchanged. (Actually, in the present one-dimensional case we may insist that all blow-ups precede all blow-downs.) As in the case $t=0$, also for each $t=1,2,\ldots,r-1,$ we have the identity $\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\Delta_{t})=\lambda_{1}(\operatorname{\mathrm{conv}}(a,b),\Delta_{t+1})$.

(iv) Let $\Delta^{\prime}$ and $\Delta^{\prime\prime}$ be the Hirzebruch-Jung desingularizations of $\operatorname{\mathrm{conv}}(a,b)$ and $(\operatorname{\mathrm{conv}}(b,c)$ respectively. Then $\Delta^{\prime}\cup\Delta^{\prime\prime}$ determines a regular complex with support $\operatorname{\mathrm{conv}}(a,c).$ By (10) and (iii),

[TABLE]

The proof is complete. ∎

Theorem 5.3** (Rational oriented segments in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry).**

Let $A=\operatorname{\mathrm{conv}}(a,b)$ be a rational oriented segment in $\mathbb{R}^{n}$. Let $x_{1}\not=a$ be the point nearest to $a$ in the Hirzebruch-Jung triangulation of $A$. Then the quadruple

[TABLE]

is a computable complete $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit invariant for $A$. If $n\not=2$, the integer $c_{\operatorname{\mathrm{aff}}(A)}$ is redundant.

Proof.

For any rational oriented segment $S$ in $\mathbb{R}^{n}$ let us write

[TABLE]

where $\operatorname{\mathrm{den}}(\mathsf{HJ}(S))=(\operatorname{\mathrm{den}}(s_{0}),\operatorname{\mathrm{den}}(s_{1}),\dots,\operatorname{\mathrm{den}}(s_{q}),\operatorname{\mathrm{den}}(s_{q+1}))$ is the sequence of the denominators of the vertices of the Hirzebruch-Jung triangulation $\mathsf{HJ}(S)$, listed in the $\sqsubseteq$-order. By Corollary 3.4, the integer $c_{\operatorname{\mathrm{aff}}(S)}$ is computable. By Proposition 5.1(iv), the map $S\mapsto$ $\mathsf{hj}(S)$ is computable.

Claim 1. For any oriented rational segment $A^{\prime}=\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ in $\mathbb{R}^{n}$ we have $\operatorname{\mathrm{conv}}(a,b)\cong\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ iff $\mathsf{hj}(A)=\mathsf{hj}(A^{\prime}).$

$(\Rightarrow)$ Suppose $\eta\colon A\cong A^{\prime}$ for some $\eta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$. By Theorem 2.5, $c_{A}=c_{A^{\prime}}$. By Proposition 5.1(i)-(ii), the two sequences of denominators in $\mathsf{HJ}(A)$ and $\mathsf{HJ}(A^{\prime})$ coincide, because $\eta$ preserves denominators and regularity. So $\mathsf{hj}(A)=\mathsf{hj}(A^{\prime}).$

$(\Leftarrow)$ Conversely, suppose $\mathsf{hj}(A)=\mathsf{hj}(A^{\prime})$. Let     $a=x_{0},\,x_{1},\dots,x_{u},x_{u+1}=b$    be the vertices of    $\mathsf{HJ}(A),$   and     $a^{\prime}=x^{\prime}_{0},\,x^{\prime}_{1},\dots,x^{\prime}_{u},x^{\prime}_{u+1}=b^{\prime}$ be the vertices of $\mathsf{HJ}(A^{\prime})$. By hypothesis, $\operatorname{\mathrm{den}}(x_{i})=\operatorname{\mathrm{den}}(x_{i}^{\prime})$ for each $i=0,\dots,u+1$. Further, $c_{\operatorname{\mathrm{aff}}(A)}=c_{\operatorname{\mathrm{aff}}(A^{\prime})}$. Since $\operatorname{\mathrm{conv}}(x_{0},x_{1})$ is regular, combining Lemma 2.1 with Theorem 3.3, we have rational points $w_{2},\dots,w_{n}$, all with the same denominator $c_{\operatorname{\mathrm{aff}}(A)}$, such that $\operatorname{\mathrm{conv}}(x_{0},x_{1},w_{2},\dots,w_{n})$ is a regular $n$-simplex in $\mathbb{R}^{n}.$ Symmetrically, Lemma 2.1 and Theorem 3.3 yield a regular $n$-simplex $\operatorname{\mathrm{conv}}(x^{\prime}_{0},x^{\prime}_{1},w^{\prime}_{2},\dots,w^{\prime}_{n})$ in $\mathbb{R}^{n}$, where $\operatorname{\mathrm{den}}(w^{\prime}_{2})=\dots=\operatorname{\mathrm{den}}(w^{\prime}_{n})=c_{\operatorname{\mathrm{aff}}(A^{\prime})}$. It follows that the denominators of the vertices of $\operatorname{\mathrm{conv}}(x^{\prime}_{0},x^{\prime}_{1},w^{\prime}_{2},\dots,w^{\prime}_{n})$ and of $\operatorname{\mathrm{conv}}(x_{0},x_{1},w_{2},\dots,w_{n})$ are pairwise equal. (Note that $d_{\operatorname{\mathrm{aff}}(A)}=d_{\operatorname{\mathrm{aff}}(A^{\prime})}$ follows from our standing hypothesis $\mathsf{hj}(A)=\mathsf{hj}(A^{\prime})$.) Iterating this construction for each $t=0,\dots,u,$ we obtain regular $n$-simplexes $\operatorname{\mathrm{conv}}(x^{\prime}_{t},x^{\prime}_{t+1},w^{\prime}_{2},\dots,w^{\prime}_{n})$ and $\operatorname{\mathrm{conv}}(x_{t},x_{t+1},w_{2},\dots,w_{n})$ such that the denominators of their vertices are pairwise equal. Thus Lemma 3.1 yields $\gamma_{t}\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ satisfying $\gamma\colon\operatorname{\mathrm{conv}}(x_{t},x_{t+1},w_{2},\dots,w_{n})\cong\operatorname{\mathrm{conv}}(x^{\prime}_{t},x^{\prime}_{t+1},w^{\prime}_{2},\dots,w^{\prime}_{n})$. The map $t\mapsto\gamma_{t}$ is computable.

We also have

[TABLE]

Indeed, let us compare $\gamma_{0}$ and $\gamma_{1}$. On the one hand, $\gamma_{0}$ maps $\operatorname{\mathrm{conv}}(x_{0},x_{1})$ onto $\operatorname{\mathrm{conv}}(x^{\prime}_{0},x^{\prime}_{1})$, and $\gamma_{1}$ maps $\operatorname{\mathrm{conv}}(x_{1},x_{2})$ onto $\operatorname{\mathrm{conv}}(x^{\prime}_{1},x^{\prime}_{2})$. On the other hand, $\operatorname{\mathrm{conv}}(x_{1},x_{2})$ is mapped onto $\operatorname{\mathrm{conv}}(x^{\prime}_{1},\gamma_{0}(x_{2}))$ by $\gamma_{0}$, and is mapped onto $\operatorname{\mathrm{conv}}(x^{\prime}_{1},x^{\prime}_{2})$ by $\gamma_{1}$. Both segments $\operatorname{\mathrm{conv}}(x^{\prime}_{1},\gamma_{0}(x_{2}))$ and $\operatorname{\mathrm{conv}}(x^{\prime}_{1},x^{\prime}_{2})$ are regular. Further, $\operatorname{\mathrm{den}}(x^{\prime}_{2})=\operatorname{\mathrm{den}}(\gamma_{1}(x_{2}))=\operatorname{\mathrm{den}}(\gamma_{0}(x_{2}))$, because $\gamma_{0}$ and $\gamma_{1}$ preserve denominators and regularity. By Lemma 4.1(i) and Proposition 5.1(i)-(ii), $\gamma_{1}(x_{2})=x^{\prime}_{2}=\gamma_{0}(x_{2}).$ Thus $\gamma_{0}$ and $\gamma_{1}$ agree over the segment $\operatorname{\mathrm{conv}}(x_{1},x_{2})$, whence they agree over all of $\mathbb{R}^{n}$, because both $\gamma_{0}$ and $\gamma_{1}$ send $w_{2},\dots,w_{n}$ to $w^{\prime}_{2},\dots,w^{\prime}_{n}$. Inductively, $\gamma_{j-1}$ agrees with $\gamma_{j}$ over the segment $\operatorname{\mathrm{conv}}(x_{j},x_{j+1})$ whence $\gamma_{j}=\gamma_{j-1}$. Thus every $\gamma_{j}$ agrees with $\gamma_{0}$, as required to settle (11) and Claim 1.

Claim 2. Let $A=\operatorname{\mathrm{conv}}(a,b)$ and $A^{\prime}=\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ be oriented rational segments in $\mathbb{R}^{n}$. Then $\operatorname{\mathrm{conv}}(a,b)\cong\operatorname{\mathrm{conv}}(a^{\prime},b^{\prime})$ iff $\mathsf{side}(A)=\mathsf{side}(A^{\prime}).$

Let $(x_{0}=a,\,\,x_{1},\dots,x_{l},\,\,x_{l+1}=b)\,\,\,\mbox{ and }\,\,\,(x^{\prime}_{0}=a^{\prime},\,\,x^{\prime}_{1},\dots,x^{\prime}_{m},\,\,x^{\prime}_{m+1}=b^{\prime})$ be the lists of vertices of the Hirzebruch-Jung desingularizations of $A$ and $A^{\prime}$. Suppose $\eta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ maps $A$ onto $A^{\prime}$. Then $c_{\operatorname{\mathrm{aff}}(A)}=c_{\operatorname{\mathrm{aff}}(A^{\prime})}$. By Claim 1($\Rightarrow$), $\mathsf{hj}(A)=\mathsf{hj}(A^{\prime})$. Then (10) yields $\lambda_{1}(A)=\lambda_{1}(A^{\prime})$. By Proposition 5.1(i), both segments $\operatorname{\mathrm{conv}}(x_{0},x_{1})$ and $\operatorname{\mathrm{conv}}(x_{0}^{\prime},x_{1}^{\prime})$ are regular. By Proposition 5.1(ii), $\eta(x_{1})=x^{\prime}_{1}$. It follows that $\mathsf{side}(A)=\mathsf{side}(A^{\prime})$.

Conversely, assume $\mathsf{side}(A)=\mathsf{side}(A^{\prime})$. We will prove

[TABLE]

By way of contradiction, assume $l<m$. Since by hypothesis $\mathsf{side}(A)=\mathsf{side}(A^{\prime})$, then $\mathsf{hj}(\operatorname{\mathrm{conv}}(a,x_{1}))=\mathsf{hj}(\operatorname{\mathrm{conv}}(a^{\prime},x_{1}^{\prime})).$ The basis in the inductive construction in Claim 1$(\Leftarrow)$ now yields a map $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $\operatorname{\mathrm{conv}}(a,x_{1})$ onto $\operatorname{\mathrm{conv}}(a^{\prime},x_{1}^{\prime}).$ By Proposition 5.1(i)-(ii), $\gamma(x_{2})=x^{\prime}_{2}.$ Inductively, $\gamma(x_{i})=x^{\prime}_{i}$ for each $i=1,\dots,l+1.$ Thus $\gamma(A)$ is a proper subset of $A^{\prime}$. By Theorem 5.2 (iv),

[TABLE]

On the other hand, since $\mathsf{side}(A)=\mathsf{side}(A^{\prime})$, from Theorem 5.2 (i) we get

[TABLE]

a contradiction that settles (12). In case $l>m$, arguing by contradiction one similarly proves (12). Thus $\gamma\colon A\cong A^{\prime}.$ The proof of Claim 2 is complete.

By Theorem 5.2(ii), the rational $\lambda_{1}(A)$ is computable. Therefore, the map $A\mapsto\mathsf{side}(A)$ is computable. The redundancy of $c_{\operatorname{\mathrm{aff}}(A)}$ for all $n\not=2$, follows from Lemma 2.4. ∎

Let $\operatorname{\mathrm{conv}}(u,v,w)$ be a rational 2-simplex in $\mathbb{R}^{n}$, with the orientation $u\to v\to w$. We say that $\operatorname{\mathrm{conv}}(u,v,w)$ is an oriented rational triangle. For any oriented rational triangle $\operatorname{\mathrm{conv}}(u^{\prime},v^{\prime},w^{\prime})$ in $\mathbb{R}^{n}$ we write $\operatorname{\mathrm{conv}}(u,v,w)\cong\operatorname{\mathrm{conv}}(u^{\prime},v^{\prime},w^{\prime})$ if there is $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ with $\gamma(u)=u^{\prime}$, $\gamma(v)=v^{\prime}$ and $\gamma(w)=w^{\prime}$, in symbols,

[TABLE]

The rational segments $\operatorname{\mathrm{conv}}(v,u)$ and $\operatorname{\mathrm{conv}}(v,w)$ determine two rational half-lines $H_{vu}\subseteq\operatorname{\mathrm{aff}}(\operatorname{\mathrm{conv}}(v,u))$ and $K_{vw}\subseteq\operatorname{\mathrm{aff}}(\operatorname{\mathrm{conv}}(v,w))$ with their common vertex $v$. We then have the (nontrivial ) angle $(H_{vu},K_{vw})\subseteq\mathbb{R}^{n}$.

The following theorem is a counterpart for $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry of the side-angle-side criterion for congruent triangles in euclidean geometry:

Theorem 5.4** (Rational oriented triangles in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry).**

A computable complete invariant of any rational oriented triangle $T=\operatorname{\mathrm{conv}}(u,v,w)\subseteq\mathbb{R}^{n}$ in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry is given by

[TABLE]

Proof.

Let $T^{\prime}=\operatorname{\mathrm{conv}}(u^{\prime},v^{\prime},w^{\prime})$ be another rational oriented triangle in $\mathbb{R}^{n}.$

If $\gamma\colon T\cong T^{\prime}$ for some $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$, then by Theorems 4.2 and 5.3, $\mathsf{tri}(T)=\mathsf{tri}(T^{\prime})$.

Conversely, suppose $\mathsf{tri}(T)=\mathsf{tri}(T^{\prime})$. Let us write for short $H=H_{vu},\,\,K=K_{vw},\,\,H^{\prime}=H_{v^{\prime}u^{\prime}},\,\,K^{\prime}=K_{v^{\prime}w^{\prime}}.$ Mimicking the proof of Theorem 4.2, we preliminarily compute the integer $c_{\operatorname{\mathrm{aff}}(T)}$, as well as the points $\mathsf{q}_{H}\in H$, $\mathsf{q}_{K}\in K$ and $\mathsf{p}_{HK}\in\operatorname{\mathrm{aff}}(T)$, and the regular triangle $R=\operatorname{\mathrm{conv}}(\mathsf{q}_{H},v,\mathsf{p}_{HK})$. Combining Lemma 2.1 and Theorem 3.3, we extend $R$ to a regular $n$-simplex $R_{*}=\operatorname{\mathrm{conv}}(\mathsf{q}_{H},v,\mathsf{p}_{HK},z_{3},\dots,z_{n})$ such that $\operatorname{\mathrm{den}}(z_{3})=\dots=\operatorname{\mathrm{den}}(z_{n})=c_{\operatorname{\mathrm{aff}}(T)}.$ We similarly extend the regular triangle $R^{\prime}=\operatorname{\mathrm{conv}}(\mathsf{q}_{H^{\prime}},v^{\prime},\mathsf{p}_{H^{\prime}K^{\prime}})$ to a regular $n$-simplex $R^{\prime}_{*}=\operatorname{\mathrm{conv}}(\mathsf{q}_{H^{\prime}},v^{\prime},\mathsf{p}_{H^{\prime}K^{\prime}},z^{\prime}_{3},\dots,z^{\prime}_{n})$ such that $\operatorname{\mathrm{den}}(z^{\prime}_{3})=\dots=\operatorname{\mathrm{den}}(z^{\prime}_{n})=c_{\operatorname{\mathrm{aff}}(T^{\prime})}.$ Since $\mathsf{angle}(H,K)=\mathsf{angle}(H^{\prime},K^{\prime})$, then $c_{\operatorname{\mathrm{aff}}(T)}=c_{\operatorname{\mathrm{aff}}(T^{\prime})}$, and both $R_{*}$ and $R^{\prime}_{*}$ are effectively computable. Since by hypothesis, $\mathsf{tri}(T)=\mathsf{tri}(T^{\prime})$, the denominators of the vertices of $R_{*}$ and $R^{\prime}_{*}$ are pairwise equal. Lemma 3.1 yields a uniquely determined map $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $R_{*}$ onto $R^{\prime}_{*}.$ It follows that $\gamma\colon R\cong R^{\prime},\mbox{ and hence }\gamma\colon H\cong H^{\prime}.$ Using Proposition 5.1(iv) we compute the two $(l+2)$-tuples

[TABLE]

listing the vertices of the Hirzebruch-Jung desingularizations $\mathsf{HJ}(\operatorname{\mathrm{conv}}(v,u))$ and $\mathsf{HJ}(\operatorname{\mathrm{conv}}(v^{\prime},u^{\prime}))$. From the hypothesis $\mathsf{side}(\operatorname{\mathrm{conv}}(v,u))=\mathsf{side}(\operatorname{\mathrm{conv}}(v^{\prime},u^{\prime}))$ it follows that $\mathsf{hj}(\operatorname{\mathrm{conv}}(v,u))=\mathsf{hj}(A\operatorname{\mathrm{conv}}(v^{\prime},u^{\prime})).$ Therefore, $l=m$ and for each $i=0,\dots,l+1$, $\gamma$ sends the $i$th vertex of $\mathsf{HJ}(\operatorname{\mathrm{conv}}(v,u))$ to the $i$th vertex of $\mathsf{HJ}(\operatorname{\mathrm{conv}}(v^{\prime},u^{\prime}))$. Thus $\gamma\colon\operatorname{\mathrm{conv}}(v,u)\cong\operatorname{\mathrm{conv}}(v^{\prime},u^{\prime}).$ The assumption $\mathsf{angle}(H,K)$ = $\mathsf{angle}(H^{\prime},K^{\prime})$ entails $\gamma\colon K\cong K^{\prime},$ whence $\gamma\colon\widehat{HK}\cong\widehat{H^{\prime}K^{\prime}}.$ From $\mathsf{side}(\operatorname{\mathrm{conv}}(v,w))=\mathsf{side}(\operatorname{\mathrm{conv}}(v^{\prime},w^{\prime}))$ it follows that $\gamma\colon\operatorname{\mathrm{conv}}(v,w)\cong\operatorname{\mathrm{conv}}(v^{\prime},w^{\prime}).$ Summing up, $\gamma\colon T\cong T^{\prime}$.

The computability of the map    $T\mapsto\mathsf{tri}(T)$   follows from Theorems 4.2 and 5.3. ∎

6. Classification of ellipses in

$\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-geometry

For notational simplicity, all ellipses in this paper are assumed to lie in $\mathbb{R}^{2}$.

Our construction of a computable complete invariant for ellipses in $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-geometry primarily rests on the following properties of conjugate diameters, recorded by Apollonius of Perga [1] and Pappus of Alexandria [21]:

Proposition 6.1**.**

Every ellipse $E\subseteq\mathbb{R}^{2}$ is the image of a circle under a contraction with respect to some line. $E$ has a unique center of symmetry. Calling a diameter of $E$ any chord passing through the center, it follows that the center of $E$ bisects any diameter. Let $C$ be a diameter of $E$. There is a uniquely determined diameter $C^{*}$ having the property that the middle points of all chords of $E$ parallel to $C$ lie in $C^{*}$. The latter is known as the conjugate diameter of $C$. We have $C^{**}=C$. Let $T$ be the tangent of $E$ at a point $x\in E$. Let $X$ be the diameter of $E$ containing $x$. Then the conjugate diameter $X^{*}$ is parallel to $T$.

Proof.

All these properties follow from the fact that every affine transformation can be represented as a composition of a similarity transformation and a contraction with respect to some line. ∎

Lemma 6.2**.**

Let $\phi(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f$ be a polynomial with rational coefficients. Then it is decidable whether the solution set (the zeroset)

[TABLE]

is an ellipse $E$ containing a rational point. Further, whenever any such point exists in $Z(\phi)$, the set of rational points in $E$ is dense in $E$, and can be recursively enumerated in the lexicographic order of increasing denominators.

Proof.

As explained, e.g., in [23, §5.2], $E$ contains a rational point iff the Legendre equation $px^{2}+qy^{2}+rz^{2}=0$ has an integer solution with $\gcd(x,y,z)=1$, for suitable integers $p,q,r$ which are effectively computable from the coefficients $a,\dots,f$. Perusal of [12, 17.3] shows that this latter problem is decidable, and whenever a solution exists it can be effectively computed. The rest is clear. (See [6] and [25] for efficient computations of rational points on rational conics.) ∎

By a rational ellipse we mean an ellipse $E\subseteq\mathbb{R}^{2}$ that coincides with the zeroset $Z(\phi)$ of a quadratic polynomial $\phi(x,y)$ with rational coefficients, and contains a rational point (equivalently, $E$ contains a dense set of rational points). We denote by $\mathcal{E}$ the set of rational ellipses. For any $E\in\mathcal{E}$, its center is a rational point. A diameter $C$ of $E$ is rational (meaning that its vertices are rational) iff so is its conjugate. Given a rational diameter $C$ in $E$, its conjugate is effectively computable. Two semi-diameters $A,B$ of $E$ are said to be conjugate iff they lie in conjugate diameters of $E$.

For any map $\gamma\in\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$, the image $E^{\prime}=\gamma(E)$ of any $E\in\mathcal{E}$ is a member of $\mathcal{E}.$ Further, $\gamma$ is a denominator preserving one-one map of all rational points of $E$ onto all rational points of $E^{\prime}$.

The proof of the following result now routinely follows from Proposition 6.1:

Lemma 6.3**.**

(i) For any pair of distinct segments $C,D$ with a common vertex and $\operatorname{\mathrm{aff}}(C)\not=\operatorname{\mathrm{aff}}(D)$, there is a unique ellipse $E$ such that $(C,D)$ is a pair of conjugate semi-diameters of $E$.333Pappus [21, Book VIII, §XVII, Proposition 14] constructs the axes of an ellipse from any given pair of conjugate semi-diameters. Further, if the segments $C$ and $D$ are rational then $E$ is a rational ellipse, which can be effectively obtained from $(C,D)$.

(ii) Two ellipses $E,E^{\prime}\in\mathcal{E}$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit iff there are conjugate rational semi-diameters $A,B$ of $E$ and $A^{\prime},B^{\prime}$ of $E^{\prime}$ such that the triangles $T=\operatorname{\mathrm{conv}}(A\cup B)$ and $T^{\prime}=\operatorname{\mathrm{conv}}(A^{\prime}\cup B^{\prime})$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit. Moreover, if $\delta\in\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ maps $T$ onto $T^{\prime}$, then $\delta$ maps $E$ onto $E^{\prime}$.

Let $O$ be the center of $E\in\mathcal{E}$. Let $(A,B)$ be a pair of conjugate rational semi-diameters of $E$, say $A=\operatorname{\mathrm{conv}}(O,x)$ and $B=\operatorname{\mathrm{conv}}(O,y)$. Then the sum of the denominators of $x$ and $y$ is said to be the index of $(A,B)$.

Theorem 6.4** (Rational ellipses in $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-geometry).**

For any $E\in\mathcal{E}$ let $\{D_{1},\dots,D_{q}\}$ be the set of all pairs $D_{i}=(A_{i},B_{i})$ of conjugate semi-diameters of $E$ having the smallest index. For each $i=1,\dots,q$ let the triangle $T_{i}=\operatorname{\mathrm{conv}}(A_{i}\cup B_{i})$ be oriented so that $O$ is the first vertex, followed by the vertex of $A_{i}$, followed by the vertex of $B_{i}$. With    $\mathsf{tri}(T_{i})$    the invariant defined in Theorem 5.4, let

[TABLE]

We then have:

• (i)

$\mathsf{ell}$* is a complete $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit invariant of ellipses in $\mathcal{E}$.*

• (ii)

For any rational quadratic polynomial $\phi(x,y)$ whose zeroset $Z(\phi)$ is an element of $\mathcal{E}$, (a decidable condition, by Lemma 6.2), the map $\phi\mapsto\mathsf{ell}(Z(\phi))$ is computable.

• (iii)

Thus there is a decision procedure for the problem whether two rational ellipses $E,E^{\prime}\in\mathcal{E}$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit. When this is the case, a map $\gamma\in\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ of $E$ onto $E^{\prime}$ can be effectively computed.

Proof.

(i) For any $E,E^{\prime}\in\mathcal{E}$ we must show:

[TABLE]

$(\Leftarrow)$ Suppose $\mathsf{ell}(E)=\mathsf{ell}(E^{\prime})$. By assumption, $E$ has a pair $(A,B)$ of rational conjugate semi-diameters of smallest index $d$,   and $E^{\prime}$ has a pair $(A^{\prime},B^{\prime})$ of rational conjugate semi-diameters of the same smallest index $d$, such that the two triangles $\operatorname{\mathrm{conv}}(A\cup B)$ and $\operatorname{\mathrm{conv}}(A^{\prime}\cup B^{\prime})$ have the same invariants. By Theorem 5.4, the two triangles $\operatorname{\mathrm{conv}}(A\cup B)$ and $\operatorname{\mathrm{conv}}(A^{\prime}\cup B^{\prime})$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit. By Lemma 6.3, $E$ and $E^{\prime}$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit.

$(\Rightarrow)$ Let $\gamma\in\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ map $E$ onto $E^{\prime}$. Let $O$ be the center of $E$, and $O^{\prime}$ the center of $E^{\prime}$. Since $\gamma$ preserves ratios of collinear segment lengths, as well as parallel and tangent lines, then by Lemma 6.3, $O^{\prime}=\gamma(O)$. Further, $\gamma$ sends any pair $(A,B)$ of conjugate semi-diameters of $E$ to a pair $(A^{\prime},B^{\prime})$ of conjugate semi-diameters of $E^{\prime}=\gamma(E)$. The preservation properties of the affine transformation $\gamma$ ensure that the image $\gamma(E)$ coincides with the ellipse constructed from $(A^{\prime},B^{\prime})$ according to Lemma 6.3. Pick a triangle $T$ of $E$ arising from a pair of semi-diameters of smallest index $d$. Since $\gamma$ preserves all numerical invariants in Theorems 4.2 and 5.3, then the two sides of $\gamma(T)$ having $O^{\prime}$ as a common vertex will be conjugate semi-diameters of $E^{\prime}$ of smallest index $=d$. Further, $\mathsf{tri}(T)=\mathsf{tri}(\gamma(T))$. It follows that $\gamma$ induces a bijection $\beta$ between pairs $(A_{j},B_{j}),\,\,\,j=1,\dots,q,$ of conjugate semi-diameters of $E$ of smallest index, and pairs $(\beta(A_{j}),\beta(B_{j}))$ of conjugate semi-diameters of $E^{\prime}$ of smallest index, and we may write $\gamma\colon\operatorname{\mathrm{conv}}(A_{j}\cup B_{j})\cong\operatorname{\mathrm{conv}}(\beta(A_{j})\cup\beta(B_{j}))$. By Theorem 5.4, $\mathsf{tri}(T_{j})=\mathsf{tri}(\gamma(T_{j}))$ for each $j=1,\dots,q$. By definition, $\mathsf{ell}(E)=\mathsf{ell}(E^{\prime})$, which completes the proof of (i).

(ii) To prove the computability of the map $\phi\mapsto\mathsf{ell}(Z(\phi))$ we preliminarily check that the zeroset $Z(\phi)$ is a member of $\mathcal{E}$. By Lemma 6.2, this condition can be decided by a Turing machine over the input given by the coefficients of $\phi$. If the condition is satisfied, letting $E=Z(\phi)$ we proceed as follows:

1. We compute the (automatically rational) center $O$ of $E$, and let $S$ be a closed square with rational vertices in $\mathbb{R}^{2}$, centered at $O$ and containing $E$;
2. For each $j=1,2,\dots$ we let $X_{j}$ be the set of rational points of $E$ of denominator $\leq j$. $X_{j}$ is effectively computable, because there are only finitely many rational points $x\in S$ of denominator $\leq j,$ and it is decidable whether any such point $x$ lies in $E$;
3. For any pair $(x,y)$ of points in $X_{j}$ we check whether $(\operatorname{\mathrm{conv}}(O,x),\operatorname{\mathrm{conv}}(O,y))$ is a pair of conjugate semi-diameters of $E$. As already noted, this can be done in an effective way;
4. Let $d$ be the smallest integer such that $X_{d}$ contains two points $x,y$ having the property that $(\operatorname{\mathrm{conv}}(O,x),\operatorname{\mathrm{conv}}(O,y))$ is a pair of rational conjugate semi-diameters of $E$ of index $d$. Since $E$ does have rational conjugate semi-diameters, after a finite number of steps such $d$ will be found;
5. Let $\{D_{1},\dots,D_{q}\}$ be the (necessarily finite) set of all pairs of conjugate semi-diameters of $E$ of index $d$. For any $D_{i}=(A_{i},B_{i})$, letting $T_{i}=\operatorname{\mathrm{conv}}(A_{i}\cup B_{i})$, we compute $\mathsf{tri}(T_{i})$ as in Theorem 5.4. We finally write $\mathsf{ell}(E)=\{\mathsf{tri}(T_{1}),\dots,\mathsf{tri}(T_{q})\}$.

Since all these steps are effective, the map $\phi\mapsto\mathsf{ell}(Z(\phi))$ is computable.

(iii) This immediately follows from the proof of (i) and (ii). The proof of (ii) also shows that there is a Turing machine having the following property: whenever $E$ and $E^{\prime}$ have the same $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-orbit, a map $\gamma\in\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ of $E$ onto $E^{\prime}$ is effectively obtainable from the input data $\phi$ and $\phi^{\prime}$. ∎

The computable complete invariant   $\mathsf{ell}$   of the foregoing theorem is here to stay, because of the following Turing equivalence result, whose proof is similar to the proof of Proposition 4.4:

Proposition 6.5** (Universal property of $\mathsf{ell}$).**

Suppose $\mathsf{newell}$ is a computable complete invariant of ellipses in $\mathcal{E}$ in $\mathsf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$-geometry. Then there is a Turing machine $\mathcal{R}$ which, over any input string $\alpha$ coinciding with $\mathsf{newell}(E)$ for some $E\in\mathcal{E}$, outputs the string $\mathcal{R}(\alpha)=\mathsf{ell}(E)$. Conversely, there is a Turing machine $\mathcal{S}$ which, over any input string $\beta=\mathsf{ell}(F)$ for some $F\in\mathcal{E}$, outputs the string $\mathcal{S}(\beta)=\mathsf{newell}(F)$. The two maps $\alpha\mapsto\mathcal{R}(\alpha)$ and $\beta\mapsto\mathcal{S}(\beta)$ are inverses of each other.

7. Polyhedra in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ geometry

Following [26, 1.1], by a polyhedron (“compact polyhedron”, in [22, 2.2]) we mean the union $P=\bigcup_{i}S_{i}$ of finitely many simplexes $S_{i}$ in $\mathbb{R}^{n}$. $P$ need not be convex or connected. The $S_{i}$ need not have the same dimension. If the vertices of each $S_{i}$ have rational coordinates, $P$ is said to be a rational polyhedron.

Rational polyhedra play a key role in the recognition problem of combinatorial manifolds presented as rational polyhedra $X,Y$. As a matter of fact, (see, e.g., [10, p.55]), $X$ is homeomorphic to $Y$ iff there is a rational PL-homeomorphism $\eta$ of $X$ onto $Y$, i.e., a finitely piecewise affine linear (PL) one-one continuous map $\phi$ of $X$ onto $Y$ such that every affine linear piece of $\phi$ has rational coefficients.

It follows that the set $\mathcal{S}$ of pairs of rationally PL-homeomorphic polyhedra is recursively enumerable. And yet, the complementary set is not:

Theorem 7.1** **(A.A. Markov, 1958.

See [5, 24] and references therein).

The problem whether two rational polyhedra $X$ and $Y$ are rationally PL-homeomorphic is undecidable.

While stated in terms of rational polyhedra, this theorem had enough impact to put an end to the (Klein) program of attaching to any combinatorial manifold $X$ an invariant characterizing $X$ up to homeomorphism.

It is an interesting open problem whether Markov unrecognizability theorem still holds when rational PL-homeomorphisms are replaced by integer PL-homeomorphisms.

Given rational polyhedra $P,P^{\prime}\subseteq\mathbb{R}^{n}$ let us agree so write $P\cong P^{\prime}$ if some $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ maps $P$ onto $P^{\prime}$, in symbols, $\gamma\colon P\cong P^{\prime}.$

Theorem 7.2** (Recognizing rational polyhedra in $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-geometry).**

The following problem is decidable:

${\mathsf{INSTANCE}:}$* Rational polyhedra $P=\bigcup_{i=1}^{l}S_{i}$ and $P^{\prime}=\bigcup_{j=1}^{m}T_{j}\,,$ where each $S_{i}$ and $T_{j}$ is a rational simplex in $\mathbb{R}^{n}$, presented by the list of its vertices.*

${\mathsf{QUESTION}:}$* Does there exist $\delta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\delta(P)=P^{\prime}$?*

Moreover, whenever such $\delta$ exists it can be effectively computed.

Proof.

From the two lists of simplexes $S_{i},T_{j}$ we construct rational triangulations $\nabla$ of $P$ and $\nabla^{\prime}$ of $P^{\prime}$ following [26, Chapter II]. (Also see [11, §§17 and 25].) From the vertices of $\nabla$ and $\nabla^{\prime}$, the algorithmic procedure of [11, §§7 and 22] yields the (vertices of the) convex hulls $C=\operatorname{\mathrm{conv}}(P)\,\,\,\mbox{ and }\,\,\,C^{\prime}=\operatorname{\mathrm{conv}}(P^{\prime})\subseteq\mathbb{R}^{n}.$ Let

[TABLE]

If $P\cong P^{\prime}$ then $C\cong C^{\prime}$ and $F\cong F^{\prime}$. Using the decision procedure of Corollary 3.4 we check whether the affine subspaces $F$ and $F^{\prime}$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit. If this condition fails, our problem has a negative answer. Otherwise, we introduce the notation

[TABLE]

Corollary 3.4 yields a map $\gamma\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that

[TABLE]

Since the rational polyhedron $C$ is $e$-dimensional and convex, we fix, once and for all, rational points $r_{0},\dots,r_{e}\in C$ and additional points $r_{e+1},\dots,r_{n}\in\mathbb{Q}^{n}$ such that $\operatorname{\mathrm{conv}}(r_{0},\dots,r_{n})$ is an $n$-simplex in $\mathbb{R}^{n}$. Using, if necessary, the desingularization procedure described in [7, VI, 8.5] or [8, p.48], we may safely assume that $\operatorname{\mathrm{conv}}(r_{0},\dots,r_{n})$ is regular. Let us use the abbreviations

[TABLE]

For each $i=0,\dots,n$ let us set

[TABLE]

Observe that $G^{*}=\gamma(R^{*})$ is a regular $n$-simplex, $G=\gamma(R)$ is a regular $e$-simplex, and   $\operatorname{\mathrm{aff}}(G)$   coincides with $F^{\prime}$ by (13).

Claim. The following conditions are equivalent:

• (I)

There exists a map $\delta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $P$ onto $P^{\prime}$.

• (II)

There are rational points $s_{0},\dots,s_{e}\in C^{\prime}$ with the following properties:

(i) $\operatorname{\mathrm{den}}(s_{i})=\operatorname{\mathrm{den}}(r_{i})$ for each $i=0,\dots,e$.

(ii) $\operatorname{\mathrm{conv}}(s_{0},\dots,s_{e})$ is a regular $e$-simplex; (Thus by Lemma 2.1 the $n$-simplex $\operatorname{\mathrm{conv}}(s_{0},\dots,s_{e},g_{e+1},\dots,g_{n})$ is regular, because $G^{*}$ is regular and $\operatorname{\mathrm{aff}}(\{s_{0},\dots,s_{e}\})=\operatorname{\mathrm{aff}}(C^{\prime})=\operatorname{\mathrm{aff}}(G)=F^{\prime}$).

(iii) Letting $W\subseteq(\mathbb{Q}^{n})^{n+1}$ be defined by

[TABLE]

the map $\phi=\phi_{VW}$ of $R^{*}$ onto $\operatorname{\mathrm{conv}}(W)$ given by Lemma 3.1 sends $P$ onto $P^{\prime}$. (Lemma 3.1 can be applied because both simplexes $R^{*}$ and $\operatorname{\mathrm{conv}}(W)$ are regular and the denominators of their vertices are pairwise equal.)

For the nontrivial direction, suppose some $\delta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ maps $P$ onto $P^{\prime}$. Then $\delta\colon C\cong C^{\prime}$ and $\delta\colon F\cong F^{\prime}.$ For each $i=0,\dots,n$ let us define the rational point $s_{i}\in C^{\prime}$ by

[TABLE]

with the intent of proving that $s_{0},\dots,s_{e}$ satisfy conditions (i)-(iii).

Condition (i) is immediately satisfied, because $\delta$ preserves denominators. Next, let us set $S=\operatorname{\mathrm{conv}}(s_{0}\dots,s_{e})=\delta(R)\mbox{ and }S^{*}=\operatorname{\mathrm{conv}}(s_{0}\dots,s_{n})=\delta(R^{*}).$ Since $S^{*}$ is regular then so is $S$, and condition (ii) is satisfied. There remains to be proved that $s_{0},\dots,s_{e}$ satisfy condition (iii). To this purpose let us first note that both $\gamma$ and $\delta$ map $F$ onto $F^{\prime}$, and hence $\operatorname{\mathrm{aff}}(S)=\operatorname{\mathrm{aff}}(G)=F^{\prime}.$ Further, from $\gamma\colon R^{*}\cong G^{*}\,\,\,\mbox{and}\,\,\,\delta\colon R^{*}\cong S^{*}$ we get $S^{*}\cong G^{*}.$ By restriction, we obtain regular $e$-simplexes $R\cong G\cong S$ having the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit. Let $S^{\leftarrow}=\operatorname{\mathrm{conv}}(s_{0},\dots,s_{e},g_{e+1},\dots,g_{n}).$ Since the vertices $g_{e+1},\dots,g_{n}$ are common to both $G^{*}$ and $S^{\leftarrow}$, by (16) we have

[TABLE]

Since $G^{*}$ is a regular $n$-simplex, by Lemma 2.1 so is $S^{\leftarrow}$. Therefore, by Lemma 3.1, there is a uniquely determined $\beta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\beta\colon G^{*}\cong S^{\leftarrow}.$ The two $n$-simplexes $S^{\leftarrow}$ and $S^{*}$ are regular and their vertices have pairwise equal denominators, because their first $e+1$ vertices $s_{0},\dots,s_{e}$ coincide, and by (17),

[TABLE]

Another application of Lemma 3.1 yields a uniquely determined $\alpha\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ such that $\alpha\colon S^{\leftarrow}\cong S^{*}.$ It follows that $\delta=\alpha\circ\beta\circ\gamma,$ where “$\circ$” denotes composition. Let $\phi=\beta\circ\gamma.$ Then $\phi\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ maps $R^{*}$ onto $S^{\leftarrow}$. Specifically, recalling (14)-(15), $\phi$ coincides with the map $\phi_{VW}$ of Lemma 3.1. By construction, $\phi$ agrees with $\delta$ over $R$ (whence $\phi$ agrees with $\delta$ over $F\supseteq C\supseteq P$). Since $\delta$ maps $P$ onto $P^{\prime}$, then so does $\phi.$ Thus the points $s_{0},\dots,s_{e}$ satisfy condition (iii). Our claim is settled.

Let now $\Omega$ be the set of all $(e+1)$-tuples $s=(s_{0},\dots,s_{e})$ of rational points in $C^{\prime}$ such that $\operatorname{\mathrm{den}}(s_{i})=\operatorname{\mathrm{den}}(r_{i})$ for all $i=0,\dots,e$, and the set $T_{s}=\operatorname{\mathrm{conv}}(s_{0},\dots,s_{e})$ is a regular $e$-simplex—a condition that can be effectively checked. $\Omega$ is a finite set, because $C^{\prime}$ is bounded and there are only finitely many rational points in $C^{\prime}$ with denominators $\leq\max(\operatorname{\mathrm{den}}(r_{0}),\dots,\operatorname{\mathrm{den}}(r_{e})).$ It is easy to see that $\Omega$ is the set of all $(e+1)$-tuples of rational points in $C^{\prime}$ satisfying conditions (i)-(ii) in our claim. Letting the $n$-simplex $T^{\leftarrow}_{s}$ be defined by $T^{\leftarrow}_{s}=\operatorname{\mathrm{conv}}(s_{0},\dots,s_{e},g_{e+1},\dots,g_{n}),$ the regularity of $T^{\leftarrow}_{s}$ follows from the regularity of $T_{s}$ and of $G^{*}$, by Lemma 2.1. From the $(n+1)$-tuple of rational points $V$ defined in (14) and the $(n+1)$-tuple $U(s)=(s_{0},\dots,s_{e},g_{e+1},\dots,g_{n}),$ Lemma 3.1 yields a uniquely determined map $\phi_{VU(s)}\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $R^{*}$ onto $T^{\leftarrow}_{s}$.

By our claim, $P\cong P^{\prime}$ iff for at least one $(e+1)$-tuple $\bar{s}=(\bar{s}_{0},\dots,\bar{s}_{e})\in\Omega$, $\phi_{VU(\bar{s})}$ maps $P$ onto $P^{\prime}$, i.e., $\bar{s}$ also satisfies condition (iii). This final condition is decidable, by resorting to the triangulations $\nabla$ and $\nabla^{\prime}$ constructed at the outset of the proof: indeed, we must only check whether for each simplex in $\nabla$ its $\phi_{VU(\bar{s})}$-image is contained in the union of simplexes of $\nabla^{\prime},$ and vice-versa, check whether for each simplex in $\nabla^{\prime}$ its $\phi_{VU(\bar{s})}^{-1}$-image is contained in the union of simplexes of $\nabla.$

We have just shown the decidability of the problem whether there is a map $\delta\in\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$ of $P$ onto $P^{\prime}$. Our constructive proof also shows that whenever any such $\delta$ exists, it can be effectively computed. ∎

Figure 2 illustrates the crux of the proof for $n=3$ and $e=2$.

Concluding Remarks

Suppose $P$ and $Q$ are finite unions of $n$-dimensional rational simplexes in $\mathbb{R}^{n}$ (for short, $P$ and $Q$ are rational “$n$-polyhedra”). Suppose there are rational triangulations $\Delta$ of $P$ and $\nabla$ of $Q$ such that every simplex $T$ of $\Delta$ can be mapped one-one onto a simplex of $\nabla$ by some $\eta_{T}\in\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$, in such a way that the set $\eta=\bigcup\{\eta_{T}\mid T\in\Delta\}$ is a continuous one-one map. We then say that $\eta$ is a “continuous $\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$-equidissection”, [11, 31.3].

Markov unrecognizability theorem is to the effect that the continuous $\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$-equidissectability of $P$ and $Q$ is not decidable. It is an interesting open problem whether Markov’s theorem still holds when continuous $\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$-equidissections are replaced by continuous $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-equidissections. The subproblem of deciding whether $P$ and $Q$ have the same $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit has been shown to be decidable in Theorem 7.2.

Differently from the case of angles, segments, triangles and ellipses, our positive solution of Problem (1) for rational polyhedra does not rest on the assignment of a computable complete invariant to every rational polyhedron $P\subseteq\mathbb{R}^{n}$. And yet, $P$ is equipped with a wealth of computable invariants for continuous $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-equidissections—well beyond the classical homeomorphism invariants given by dimension, number of connected components, or Euler characteristic. These invariants include: The number of rational points in $P$ of a given denominator $d=1,2,\dots$; The number of regular triangulations $\Delta$ of $P$ such that the denominators of all vertices of $\Delta$ are $\leq d;$    The smallest possible number of $k$-simplexes in a regular triangulation $\Delta$ of $P$ such that the denominators of all vertices of $\Delta$ are $\leq d$. All these new invariants are (a fortiori), $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit invariants—and none makes sense in euclidean geometry, or even in $\mathsf{GL}(n,\mathbb{Q})\ltimes\mathbb{Q}^{n}$-geometry. Closing a circle of ideas, one may then naturally ask the following question:

Problem ($n=2,3,\dots$)

Can the $\mathsf{GL}(n,\mathbb{Z})\ltimes\mathbb{Z}^{n}$-orbit problem for rational $n$-polyhedra be decided by computable complete invariants?