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Discrete monodromy, pentagrams, and the method of condensation. (English) Zbl 1148.51001

Summary: This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly \(n\) algebraically independent invariants for the map, when it is defined on the space of \(n\)-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence.

MSC:

51A05 General theory of linear incidence geometry and projective geometries
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
52A10 Convex sets in \(2\) dimensions (including convex curves)

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