k阶线性递归模型（p）
Degree k Linear Recursions Mod(p)
摘要：
通过在整数向量$(t_1,...,t_k)$上计算广义斐波那契多项式序列$\{F_{k,n}(t_1,...,t_k)\}$（完全对称多项式的等压反射）来确定次数$k$的线性递归。如果$F_{k,n}(t_1,...,t_k) = f_n$，则$$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$和$\{f_n\}$是次数为$k$的线性递归。一方面，讨论了这类序列模素数$p$的周期性，并证明其与某些代数数域的素数结构有关，例如，周期的算术性质可用来刻画素数在扩域中的分歧.另一方面，证明了与数域相关联的半环的结构完全由Schur-hook多项式决定.
英文摘要：
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.
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