Immersion and darboux polynomials of boolean networks with application to the pseudomonas syringae hrp regulon

Boolean networks are treated in this paper as discrete-time polynomial systems over the Galois field F₂. It is shown that a previous result, valid for standard polynomial nonlinear systems, and allowing to compute the Darboux polynomials of the system, holds for polynomial systems over F₂ in a reinforced manner. Moreover, it is shown that, by using a state immersion, polynomial systems over F₂ can be always linearized. The state immersion technique can be used also for other purposes, such as to compute the solution in closed form and to compute some of the Darboux polynomials.