The pentagram map was introduced by Schwartz in 1992 as a dynamical system on polygons in the real projective plane. The map sends a polygon to the polygon formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning that it acts like a translation on a family of real tori. The concept of discrete integrability can also be framed in terms of algebraic geometry, so we can ask about integrability over any base field. Soloviev proved complex integrability in 2013. We show that the pentagram map is integrable over any algebraically closed field, except possibly in characteristic 2 and 3. This allows us to describe the dynamics of the pentagram map over finite fields.
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