Ritt分解定理的有理函数推广的反例构造
Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem
摘要：
经典的Ritt定理给出了一元多项式分解的若干性质。在本文中，我们提出了新的反例Ritt的第一定理，其中规定的长度相等的分解链的多项式，在有理函数的情况下。也就是说，我们提供了一个明确的例子，一个有理函数的系数在Q和两个不同长度的分解。另一个方面是使用一些技术，可以允许其他反例，即，相关的群体和分解，并使用的事实，即交替群A_4有两个不同长度的子群链;我们提供更多的信息，推广的另一个属性多项式分解：稳定性的基本领域。给出了一个计算有理函数的固定群的算法，该算法的复杂度为Q。
英文摘要：
The classical Ritt's Theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to Ritt's first theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in Q and two decompositions of different length. Another aspect is the use of some techniques that could allow for other counterexamples, namely, relating groups and decompositions and using the fact that the alternating group A_4 has two subgroup chains of different lengths; and we provide more information about the generalizations of another property of polynomial decomposition: the stability of the base field. We also present an algorithm for computing the fixing group of a rational function providing the complexity over Q.
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