Geodesic
Ring of Simple Polytopes and Differential Equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 18-43

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Simple polytopes are a classical object of convex geometry. They play a key role in many modern fields of research, such as algebraic and symplectic geometry, toric topology, enumerative combinatorics, and mathematical physics. In this paper, the results of a new approach based on a differential ring of simple polytopes are described. This approach allows one to apply the theory of differential equations to the study of combinatorial invariants of simple polytopes. 
@article{TM_2008_263_a2,
     author = {V. M. Buchstaber},
     title = {Ring of {Simple} {Polytopes} and {Differential} {Equations}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {18--43},
     publisher = {mathdoc},
     volume = {263},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_263_a2/}
}
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V. M. Buchstaber. Ring of Simple Polytopes and Differential Equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 18-43. http://geodesic.mathdoc.fr/item/TM_2008_263_a2/