We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao’s method for constructing elements in the finite field $${\mathbb {F}}_{q^n}$$ F q n whose orders are larger than any polynomial in n when n becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.

中文翻译：

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关于有理函数迭代的乘法独立性

我们给出了一般域上有理函数迭代（有某些例外）的乘法组合程度的下界，建立了所述迭代的乘法独立性。这导致了 Gao 在有限域 $${\mathbb {F}}_{q^n}$$ F qn 中构造元素的方法的推广，当 n 变大时，其阶数大于 n 中的任何多项式。此外，我们讨论多项式的有限性，这些多项式将给定的有限多项式集转化为乘法相关。