An improved version of the singularity-induced bifurcation theorem

It has been shown recently that there is a new type of codimension one bifurcation, called the singularity-induced bifurcation (SIB), arising in parameter dependent differential-algebraic equations (DAEs) of the form x˙=f and 0=g, and which occurs generically when an equilibrium path of the DAE crosses the singular surface defined by g=0 and det g_y=0. The SIB refers to a stability change of the DAE owing to some eigenvalue of a related linearization diverging to infinity when the Jacobian g_y is singular. In this article an improved version (Theorem 1.1) of the SIB theorem with its simple proof is given, based on a decomposition theorem (Theorem 2.1) of parameter dependent polynomials