  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.
