Combinatorics of simple polytopes and differential equations

Simple polytopes play important role in applications of algebraic geometry to physics. They are also main objects in toric topology. There is a commutative associative ring P generated by simple polytopes. The ring P possesses a natural derivation d, which comes from the boundary operator. We shall describe a ring homomorphism from the ring P to the ring of polynomials Z[t,α] transforming the operator d to the partial derivative ∂/∂t. This result opens way to a relation between polytopes and differential equations. As it has turned out, certain important series of polytopes (including some recently discovered) lead to fundamental nonlinear differential equations in partial derivative