  In this paper we define, independent of theories, two discriminant matrices
involving a solution to the scattering equations in four dimensions, the ranks
of which are used to divide the solution set into a disjoint union of subsets.
We further demonstrate, {entirely within the Cachazo-He-Yuan formalism,} that
each subset of solutions gives nonzero contribution to tree-level
$\text{N}^{k}\text{MHV}$ gauge/gravity amplitudes only for a specific value of
$k$. Thus the solutions can be characterized by the rank of their discriminant
matrices, which in turn determines the value of $k$ of the $\text{N}^{k}
\text{MHV}$ amplitudes a solution can support. As another application of the
technique developed, we show analytically that in Einstein-Yang-Mills theory,
if all gluons have the same helicity, the tree-level single-trace amplitudes
must vanish.
