The determinant of the stress matrix and restorability of hinged frameworks from self-stresses

We consider static-geometrical properties of planar hinged frameworks some of whose hinges are fastened. For such frameworks we establish some properties of the determinant of the stress matrix, including a necessary and sufficient condition for the irreducibility of this determinant as a polynomial. Using the irreducibility property, we prove the existence of uncancellable completely stressed frameworks with non-collinear fastened hinges for two infinite sequences of structure schemes. We give examples of frameworks such that the positions of all their free hinges can be restored knowing the space of self-stresses and the positions of the fastened hinges. However, this cannot be done knowing only one arbitrary stress instead of the whole space of self-stresses.