Mixed volumes of networks with binomial steady-states

The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of the steady-state system or an equivalent system. In this work, we show that for partionable binomial networks, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. We then turn our attention to identifying cycles with binomial steady-state ideals. To this end, we give a coloring condition on directed cycles that guarantees the network has a binomial steady-state ideal. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.