# zbMATH — the first resource for mathematics

On inverse permutation polynomials. (English) Zbl 1183.11075
Summary: We give an explicit formula of the inverse polynomial of a permutation polynomial of the form $$x^rf(x^s)$$ over a finite field $$\mathbb F_q$$ where $$s|q - 1$$. This generalizes results in [A. Muratović-Ribić, Finite Fields Appl. 13, No. 4, 977–980 (2007; Zbl 1167.11044)] where $$s=1$$ or $$f = g^{\frac{q-1}{s}}$$ were considered respectively. We also apply our result to several interesting classes of permutation polynomials.

##### MSC:
 11T06 Polynomials over finite fields
Full Text:
##### References:
 [1] A. Akbary, D. Ghioca, Q. Wang, On permutation polynomials of prescribed shape, Finite Fields Appl. (2009), doi:10.1016/j.ffa.2008.12.001, in press · Zbl 1220.11145 [2] Akbary, A.; Wang, Q., On some permutation polynomials, Int. J. math. math. sci., 16, 2631-2640, (2005) · Zbl 1092.11046 [3] Akbary, A.; Wang, Q., A generalized Lucas sequence and permutation binomials, Proc. amer. math. soc., 134, 1, 15-22, (2006) · Zbl 1137.11355 [4] Akbary, A.; Wang, Q., On polynomials of the form $$x^r f(x^{(q - 1) / l})$$, Int. J. math. math. sci., 7, (2007), Art. ID 23408 · Zbl 1135.11341 [5] Mullen, G.L., Permutation polynomials over finite fields, (), 131-151 · Zbl 0808.11069 [6] Muratović-Ribić, A., A note on the coefficients of inverse polynomials, Finite fields appl., 13, 4, 977-980, (2007) · Zbl 1167.11044 [7] A. Muratović-Ribić, Inverse of some classes of permutation binomials, Discrete Appl. Math., in press [8] Wan, D.; Lidl, R., Permutation polynomials of the form $$x^r f(x^{(q - 1) / d})$$ and their group structure, Monatsh. math., 112, 149-163, (1991) · Zbl 0737.11040 [9] Wang, Q., Cyclotomic mapping permutation polynomials, (), 119-128 · Zbl 1154.11342
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.