Hidden equations of risk critical thresholds

We consider the problem of parametric sensitivity of a particular characterization of risk, with respect to a threshold parameter (Formula presented.) Such threshold risk is modeled as the probability of a (Formula presented.) perturbed function of a random variable falling below 0. We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as δ approaches them. Under weak conditions, we characterize candidates for risk critical points as zeroes of either the discriminant of a relevant (Formula presented.) perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in δ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as "hidden equations of risk critical thresholds".