科研之友
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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) - Sigma(k)(j=1) 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 semi-local rings associated with the number field is shown to be completely determined by Schurhook polynomials.
- 出版日期2011
