科研之友
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摘要
Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree delta >= 2 is said to be disintegrated if it is not linearly conjugate to x(delta) or +/- T-delta(x), where T-delta(x) is the Chebyshev polynomial of degree delta. Let d and n be integers >1, we prove that there exists an effectively computable constant c(d, n) depending only on d and n such that the following holds. Let f(1), ..., f(n) is an element of K[x] be disintegrated polynomials of degree at most d and let phi = f(1) x ... x f(n) be the induced coordinate-wise self-map of A(K)(n). Then the period of every irreducible phi-periodic subvariety of A(K)(n) with non-constant projection to each factor A(K)(1) is at most c(d, n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are the Medvedev-Scanlon classification of invariant sub-varieties together with classical and more recent results in Ritt's theory of polynomial decomposition.
- 出版日期2017-1
