Abstract
This paper deals with two questions concerning permutation polynomials in several variables. Lidl and Niederreiter have considered the problem of when a sum of permutation polynomials in disjoint sets of variables is itself a permutation polynomial, and in the case of prime fields have shown that it is necessary and sufficient that at least one summand be a permutation polynomial. They also showed that in the case of non-prime fields this condition is not necessary. In this paper, a necessary and sufficient condition is obtained for the general case which specialises to the previous result for prime fields. The second part extends a criterion of Niederreiter for permutation polynomials over prime fields to any finite field.
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References
Lidl, R., Niederreiter, H.: On orthogonal systems and permutation polynomials in several variables. Acta Arith.22, 257–265 (1973)
Lidl, R., Niederreiter, H.: Finite fields. Reading, MA: Addison-Wesley 1983
McDonald, B.: Finite rings with identity. New York: Dekker, 1974
Niederreiter, H.: Permutation polynomials in several variables. Acta Sci. Math. Szeged33, 53–58 (1972)
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Matthews, R. Some results on permutation polynomials over finite fields. AAECC 3, 63–65 (1992). https://doi.org/10.1007/BF01189024
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DOI: https://doi.org/10.1007/BF01189024