The algebraic dynamics of the pentagram map

Abstract The pentagram map, introduced by Schwartz [The map. Exp. Math. 1 (1) (1992), 71–81], is a dynamical system on the moduli space of polygons in projective plane. Its real and complex dynamics have been explored detail. We study map over an arbitrary algebraically closed field characteristic not equal to 2. prove that twisted discrete integrable system, sense algebraic complete integrability: birational self-map family abelian varieties. This generalizes Soloviev’s proof integrability [F. Soloviev. Integrability Duke J. 162 (15) (2013), 2815–2853]. In course proof, we construct n -gons, derive formulas for calculate Lax representation characteristic-independent methods.