On The Morse Ensemble Polynomial Of Simplicial Complexes

We introduce the \emph{Morse ensemble polynomial} ${\text{ME}}_K(z_0,\dots ,z_d)$ of a finite simplicial complex $K$, defined as the generating function ${\text{ME}}_K={\sum }_M{\prod }_i{z}_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete critical-vector distribution over all acyclic matchings, equivalently over all discrete gradient vector fields arising from discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. In dimension one, this invariant recovers the Chari–Joswig graph formula for the $f$-vector of the discrete Morse complex in a two-variable Morse-vector normalization: ${\text{ME}}_G=z_1^{m-n}\det (z_0{z}_1{I}_n+L_G)$ for a connected graph $G$. The main new contributions are higher-dimensional and structural. First, we prove a Top-Face Recursion for adding a top-dimensional simplex, with a non-liftable correction term $F(K,\sigma ,\tau )$. The vanishing and leading obstruction of this correction term are controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, a leading obstruction layer is governed by shortest obstruction paths, and a tree-like (incidence-forest) regime of the top incidence structure gives a correction-free higher-dimensional recursion, including stacked balls as a concrete class. Second, we introduce the independence ME polynomial $\mathrm{\Phi }(G):={\text{ME}}_{\mathrm{I}\mathrm{n}\mathrm{d}(G)}$, a graph invariant which strictly refines the graph-level Morse ensemble, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{I}\mathrm{n}\mathrm{d}(G)$ through coefficients such as $[z_0]\mathrm{\Phi }(G)$.