In this work we consider systems of polynomial equations and study under what conditions the semi-algebraic set of positive real solutions is contained in a parallel translate of a coordinate hyperplane. To this end we make use of algebraic and geometric tools to relate the local and global structure of the set of positive points. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi=c, and provide a criterion to identify it. We consider the global property, namely that the whole positive part of the variety is contained in a hyperplane of the form xi=c, termed absolute concentration robustness (ACR). We show that zero sensitivity implies ACR, and identify when the two properties do not agree, via an intermediate property we term local ACR.
The motivation of this work stems from the study of robustness in biochemical systems modelling the concentration of species in a reaction network, where the terms ACR and zero sensitivity are both used to this end. Here we clarify and formalise the relation between the two approaches, and, as a consequence, we obtain a practical criterion to decide upon (local) ACR under some mild assumptions.