Finding steady-state solutions for ODE systems of zero, first and homogeneous second-order chemical reactions is NP-hard

In the context of modeling of cell signaling pathways, a relevant step is finding steady-state solutions for ODE systems that describe the kinetics of a set of chemical reactions, especially sets composed of zero, first, and second-order reactions. To compute a steady-state solution, one must set the left-hand side of each ODE as zero, hence obtaining a system of non-negative, quadratic polynomial equations. If all second-order reactions are homogeneous in respect to their reactants, then the obtained quadratic polynomial equation system will also have univariate monomials. Although it is a well-known fact that finding a root of a quadratic polynomial equation system is a NP-hard problem, it is not so easy to find a readily available proof of NP-hardness for special cases like the aforementioned one. Therefore, we provide here a self-contained proof that finding a root of non-negative, with univariate monomials quadratic polynomial equation system (NUMQ-PES) is NP-hard. This result implies that finding steady-state solutions for ODE systems of zero, first and homogeneous second-order chemical reactions is a NP-hard problem; hence, it is not a feasible approach to approximate non-homogeneous second-order reactions into homogeneous ones.