# zbMATH — the first resource for mathematics

Enumerating permutation polynomials. (English) Zbl 1401.11155
Summary: We consider the problem of enumerating polynomials over $$\mathbb{F}_q$$, that have certain coefficients prescribed to given values and permute certain substructures of $$\mathbb{F}_q$$. In particular, we are interested in the group of $$N$$-th roots of unity and in the submodules of $$\mathbb{F}_q$$. We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in [S. Konyagin and F. Pappalardi, Finite Fields Appl. 12, No. 1, 26–37 (2006; Zbl 1163.11350)]. As a consequence, we prove conditions that ensure the existence of low-degree permutation polynomials of the mentioned substructures of $$\mathbb{F}_q$$.
##### MSC:
 11T06 Polynomials over finite fields
##### Keywords:
finite fields; permutation polynomials
Full Text:
##### References:
 [1] Akbary, A.; Ghioca, D.; Wang, Q., On permutation polynomials with prescribed shape, Finite Fields Appl., 15, 195-206, (2009) · Zbl 1220.11145 [2] Akbary, A.; Ghioca, D.; Wang, Q., On constructing permutations of finite fields, Finite Fields Appl., 17, 51-67, (2011) · Zbl 1281.11102 [3] Carlitz, L., Permutations in a finite field, Proc. Am. Math. Soc., 4, 538, (1953) · Zbl 0052.03704 [4] Coulter, R.; Henderson, M.; Matthews, R., A note on constructing permutation polynomials, Finite Fields Appl., 15, 553-557, (2009) · Zbl 1215.11112 [5] Dickson, L., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math., 11, 65-120, 161-183, (1897) · JFM 28.0135.03 [6] Hermite, C., Sur LES fonctions de sept lettres, C. R. Acad. Sci. Paris, 57, 750-757, (1863) [7] Konyagin, S.; Pappalardi, F., Enumerating permutation polynomials over finite fields by degree, Finite Fields Appl., 8, 4, 548-553, (2002) · Zbl 1029.11067 [8] Konyagin, S.; Pappalardi, F., Enumerating permutation polynomials over finite fields by degree II, Finite Fields Appl., 12, 1, 26-37, (2006) · Zbl 1163.11350 [9] Konyagin, S.; Shparlinski, I., Character sums with exponential functions and their applications, (2004), Cambridge University Press Cambridge · Zbl 0933.11001 [10] Lidl, R.; Niederreiter, H., Finite fields, (1983), Addison-Wesley Reading, Mass. [11] Roman, S., Advanced linear algebra, (2008), Springer-Verlag New York · Zbl 1132.15002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.