Geodesic
Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups
Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 481-519

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The first main result of the paper is a criterion for a partially commutative group $\mathbb{G}$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb{G}$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb{G}$ (of a coordinate group over $\mathbb{G}$ ) to the elementary theories of the direct factors of $\mathbb{G}$ (to the elementary theory of coordinate groups of irreducible algebraic sets).Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb{H}$ . Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb{H}$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb{H}$ to $\mathbb{H}\,*\,F$ , where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.
DOI : 10.4153/CJM-2010-035-5
Mots-clés : 20F10, 03C10, 20F06 
Casals-Ruiz, Montserrat. Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups. Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 481-519. doi: 10.4153/CJM-2010-035-5
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     author = {Casals-Ruiz, Montserrat},
     title = {Elements of {Algebraic} {Geometry} and the {Positive} {Theory} of {Partially} {Commutative} {Groups}},
     journal = {Canadian journal of mathematics},
     pages = {481--519},
     year = {2010},
     volume = {62},
     number = {3},
     doi = {10.4153/CJM-2010-035-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-035-5/}
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