On Sequences, Rational Functions and Decomposition

TL;DR

This paper unifies and extends results on approximating sequence generating functions with rational functions, emphasizing the role of numerators, and generalizes classical theorems to broader classes of sequences and rational functions.

Contribution

It revisits and generalizes Niederreiter's theorem on linear complexities and minimal polynomials, introducing new decomposition methods for numerators and denominators in rational approximations.

Findings

Generalized Niederreiter's theorem to broader rational functions.

Developed unique decomposition methods for numerators and denominators.

Extended results to sequences with jumps in linear complexity.

Abstract

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.