Sequences generated by polynomials over integral domain

Abstract

In the first part of this dissertation, let D be an integral domain. For sequences ā = (a1, a2,, an) and I = (i1, 2,..., in) in D" with distinct i,, call ā a (D", I)-polynomial sequence if there exists f(x) € D[x] such that f(i;) = a; foe all 1 ≤ j ≤n. Criteria for a sequence to be a (D", I)-polynomial sequence are established, and explicit structures of D/P, are determined. In the second part of this dissertation, let f(x) € Z[x], call Aff(x) = f(x + 1) − f(x) a difference polynomial of f(x). Let c = (c1, c2,..., Cn-1) in Zn-1. If there exists f(x) € Z[x] such that AF ƒ (i) = c; for all 1 ≤ i ≤ n − 1, then we call c, a difference polynomial sequence of length n - 1. Denote by AP, the set of all difference polynomial sequences. Criteria for a difference polynomial sequences are established, and explicit structures of Zn-1/AP and P-1/AP are determined. In the third part of this dissertation, let D be an integral domain, I = (i1, i2,..., in) Є D" with i; it if j k and A = (( a, a,..., a1), (a2, a, a,)... (aaa)) where a, a,..., a1, a2, az a22,..., an, an,..., ar are elements in D. If there exists f(x) in D[x] such that f(m) (i) = a for all 1 ≤ j ≤ n and 0 < m <r, where f(m) (i;) = a denotes the m(th) derivative of f(x) evaluated at the point i;, call a differential polynomial sequence of length n and order (71, 72,...,n) with respect to I. Criteria for a sequence to be a differential polynomial sequence of length n and order (r1, T2,...,n) with respect to I. We also investigate the case where r; = k for all j and (n, k) = (1, k), (2, 1), (3, 1) and (2, 2).