We introduce an algebraicity criterion. It has the following form: Consider an analytic subvariety of some algebraic variety X over a global field K. Under certain conditions, if X contains many K-points, then X is algebraic over K. This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combining this criterion and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we study the invariant subvarieties for products of endomorphisms. In particular, we partially generalize Medvedev–Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on
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Original Article
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Peking Mathematical Journal 2024, 7 (1): 345-398
Published: 29 March 2023
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