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Enumerating permutation polynomials. II: $$k$$-cycles with minimal degree. (English) Zbl 1035.11062
Every permutation $$\sigma$$ on the elements of the finite field $$\mathbb F_q$$ ($$q>2$$) is uniquely represented by a polynomial $$f_\sigma\in\mathbb F_q[x]$$ of degree $$\partial(f_\sigma)\leq q-2$$. A lower bound for $$\partial(f_\sigma)$$ is given by the number of the fixed points of $$\sigma$$ ($$\sigma\not=\text{id}$$). In [Finite Fields Appl. 8, 531–547 (2002; Zbl 1029.11068)], the authors considered the problem of enumerating conjugated permutations on $$\mathbb F_q$$ whose corresponding polynomials are of degree $$<q-2$$. In this sequel, they turn particularly their attention to permutation polynomials with minimal degree.
Let $$m_{[k]}(q)$$ denote the number of $$k$$-cycles $$\sigma$$ on $$\mathbb F_q$$ for which $$\partial(f_\sigma)=q-k$$, or equivalently, $$\sum_{c\in\mathbb F_q} c^i(c-\sigma(c))=0$$ for $$i=1,\ldots,k-2$$. The authors give the upper bound $m_{[k]}(q)\leq\frac{(k-1)!}{k}\;q(q-1)$ if $$\text{ char}(\mathbb F_q)>e^{(k-3)/e}$$ and the lower bound $m_{[k]}(q)\geq\frac{\varphi(k)}{k}\;q(q-1)$ for $$q\equiv1\bmod k$$ where $$\varphi$$ denotes the Euler totient function. Their proof is based on the relation of $$m_{[k]}(q)$$ to the number of solutions in $$\mathbb F_q^{k-2}$$ of a system of equations defined over $$\mathbb Z$$. The paper concludes with the computation of $$m_{[4]}(q)$$ and $$m_{[5]}(q)$$ (and $$m_{[6]}(q)$$ in parts) by means of computer algebra.

##### MSC:
 11T06 Polynomials over finite fields
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##### References:
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