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Preprint Number 543

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543. Thomas Scanlon and Yu Yasufuku
Exponential-polynomial equations and dynamical return sets

Submission date: 11 December 2012


We show that for each finite sequence of algebraic integers α_1,...,α_n and polynomials P_1(x_1,...,x_n;y_1,...,y_n),..., P_r(x_1,...,x_n;y_1,...,y_n) with algebraic integer coefficients, there are a natural number N, n commuting endomorphisms Φ_i:G_m^N \to G_m^N of the N-th Cartesian power of the multiplicative group, a point P in G_m^N(Q), and an algebraic subgroup G \leq G_m^N so that the return set {(l_1,...,l_n) in N^n : Φ_1^{\circ l_1} \circ... \circ Φ_n^{\circ l_n}(P) in G(Q) } is identical to the set of solutions to the given exponential-polynomial equation: {(l_1,...,l_n) in N^n : P_1(l_1,...,l_n;α_1^{l_1},...,α_n^{l_n}) = ... = P_r(l_1,...,l_n;α_1^{l_1},...,α_n^{l_n}) = 0 }.

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Full text arXiv 1212.1836: pdf, ps.

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