
Methods and ModelsMethods and Models 37
2.6.3 Differential Ring of Polytopes
Another method correlating the polytopes to PDE was developed by
Buchstaber (Buchstaber 2008).
A polytope P
n
of dimension n is said to be simple if every vertex of P
is the intersection of exactly n facets, that is, facets of dimension n–1. The
collection of all n-dimensional combinatorial simple polytopes is denoted
by Pn. Let P =
∑
≥0n
Pn
.
The approach is based on the differential ring, P, of combinatorial
polytopes. The ring P possesses a natural derivation d: P→P.
This allows applying the theory of differential equations to the study
of polytopes and describing generating ...