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Polynomials over finite fields with the commuting diagram properties. (Chinese. English summary) Zbl 0919.11085

Summary: We give explicit expressions for the general solutions of the polynomials \(f(x)\) over finite field \(F_q\) satisfying the condition \(f(g(x))\equiv h(f(x))\) and give the counting formula for the number of such \(f(x)\) with \(\deg f<q\), where \(g(x)\) and \(h(x)\) are two given polynomials over \(F_q\) and one of them is a permutation polynomial. This generalizes the main results in [C. Y. Chao, J. Algebra 163, 295-311 (1994; Zbl 0799.11057)] and also generalizes the main results in [C. Wells, Proc. Am. Math. Soc. 46, 347-350 (1974; Zbl 0298.12009) and G. L. Mullen, ibid. 84, 315-317 (1982; Zbl 0498.12018)] in the special cases where \(g(x)\) and \(h(x)\) are both linear polynomials.

MSC:

11T06 Polynomials over finite fields
12E05 Polynomials in general fields (irreducibility, etc.)
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