Fans, decision problems and generators of free abelian ℓ\ell-groups

Abstract

Abstract Let t 1 , … , t n {t_{1},\ldots,t_{n}} be ℓ {\ell} -group terms in the variables X 1 , … , X m {X_{1},\ldots,X_{m}} . Let t ^ 1 , … , t ^ n {\hat{t}_{1},\ldots,\hat{t}_{n}} be their associated piecewise homogeneous linear functions. Let G be the ℓ {\ell} -group generated by t ^ 1 , … , t ^ n {\hat{t}_{1},\ldots,\hat{t}_{n}} in the free m-generator ℓ {\ell} -group 𝒜 m {\mathcal{A}_{m}} . We prove: (i) the problem whether G is ℓ {\ell} -isomorphic to 𝒜 n {\mathcal{A}_{n}} is decidable; (ii) the problem whether G is ℓ {\ell} -isomorphic to 𝒜 l {\mathcal{A}_{l}} (l arbitrary) is undecidable; (iii) for m = n {m=n} , the problem whether { t ^ 1 , … , t ^ n } {\{\hat{t}_{1},\ldots,\hat{t}_{n}\}} is a 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 ℓ {\ell} -group G. We make pervasive use of fans and their stellar subdivisions.