Degree-k linear recursions mod(p) and number fields

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 related to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period are shown to characterize ramification of primes in an extension field. On the other hand, the structure of the semilocal rings associated with the number field is shown to be completely determined by Schur-hook polynomials. Keywords: Symmetric polynomials, Schur polynomials, linear recursions, number fields.