关于数列、有理函数和分解
On Sequences, Rational Functions and Decomposition
摘要：
我们的总体目标是统一和推广文献中的一些结果，在一个域上的有限和无限序列的生成函数的逼近有理函数。在我们的方法中，分子起着重要的作用。我们重温一个定理Niederreiter（一）线性复杂性和（二）“$n^{th}$最小多项式”的一个无限序列，证明了使用部分的多项式。我们证明（一）和它的逆从第一原则和推广（二）到合理的功能，分母不需要有最小的程度。我们证明（ii）在两个部分：首先是几何序列，然后在线性复杂度的跳跃序列。基本思想是将分母分解为两个最小次数多项式的多项式倍数之和;分子也有类似的分解。当分母的度数最多为序列的长度时，分解是唯一的。证明也适用于合理的功能有关的有限序列，概括的结果梅西。我们给出了一些与序列相关的有理函数的应用。
英文摘要：
Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
