Degree k Linear Recursions Mod(p)

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.