Fans and generators of free abelian l-groups

Abstract

Let $t_1,\dots ,t_n$ be $\ell$-group terms in the variables $X_1,\dots ,X_m$. Let ${\hat{t}}_1,\dots ,{\hat{t}}_n$ be their associated piecewise homogeneous linear functions. Let $G$ be the $\ell$-group generated by ${\hat{t}}_1,\dots ,{\hat{t}}_n$ in the free $m$-generator $\ell$-group ${\mathcal{A}}_m.$ We prove: (i) the problem whether $G$ is $\ell$-isomorphic to ${\mathcal{A}}_n$ is decidable; (ii) the problem whether $G$ is $\ell$-isomorphic to ${\mathcal{A}}_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether $\{{\hat{t}}_1,\dots ,{\hat{t}}_n\}$ is a {\it free} generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the $\ell$-group $G$. We make pervasive use of fans and their stellar subdivisions.

arXiv comment12 pages