Degree k Linear Recursions Mod(p)

TL;DR

This paper explores the properties of degree k linear recursions modulo prime p, linking their periodicity to algebraic number field structures and prime ramification, using generalized Fibonacci polynomials and Schur-hook polynomials.

Contribution

It introduces a novel connection between linear recursions modulo p and algebraic number theory, particularly prime ramification and semilocal ring structures.

Findings

Periodic properties relate to prime ramification in number fields

Semilocal ring structures are characterized by Schur-hook polynomials

Arithmetic of periods characterizes prime ramification

Abstract

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\{f_n\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be rela ted to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period ar e shown to characterize ramification of primes in an extension field. On the other hand, the structure of the semiloca l rings associated with the number field is shown to be completely determined by Schur-hook polynomials.