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Bases for integer-valued polynomials in a Galois field. (English) Zbl 0923.11164
Let \(GF[q,x]\) be the ring of polynomials over the Galois field \(GF(q)\) of characteristic \(p\) with \(q=p^n\) and \(GF(q,x)\) its quotient field. An integer-valued polynomial is a polynomial \(f(T)\in GF(q,x)[T]\) such that \(f(M)\in GF[q,x]\) for all \(M\in GF[q,x]\). Denote by \(D^r\), \(r\in \mathbb{N}\), respectively \(D^\infty\) the class of integer-valued polynomials which together with their derivatives up to order \(r\), respectively of all orders, are integer-valued. Denote also by \(\Delta^r\), \(r\in\mathbb{N}\), respectively \(\Delta^\infty\), the class of integer-valued polynomials which together with their differences up to order \(r\), respectively of all orders, are integer-valued. Bases for \(\Delta^r\) and \(\Delta^\infty\) over \(GF[q,x]\) are known. The author gives bases for \(D^1\) and \(D^2\) over \(GF[q,x]\) and proves that for a positive integer \(r\), we have \(\Delta^r\subset D^r\) and \(\Delta^\infty= D^\infty\).

MSC:
11T06 Polynomials over finite fields
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11C08 Polynomials in number theory
11T55 Arithmetic theory of polynomial rings over finite fields
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