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Enumerating permutation polynomials. I: Permutations with non-maximal degree. (English) Zbl 1029.11068
Every permutation $$\sigma$$ on the elements of $$\mathbb F_q$$ ($$q>2$$) is uniquely represented by a polynomial $$f_\sigma\in\mathbb F_q[x]$$ of degree $$\leq q-2$$. A lower bound for the degree of $$f_\sigma$$ is given by the number of fixed points of $$\sigma$$ ($$\sigma\not=\text{id}$$). The authors deal with the problem of enumerating conjugated permutations of $$\mathbb F_q$$ whose corresponding polynomials are of degree $$<q-2$$. This restriction is equivalent to $$\sum_{c\in\mathbb F_q} c\sigma(c)=0$$.
Let $$N_{\mathcal C}(q)$$ denote the number of permutations of $$\mathbb F_q$$ which are of cycle type $${\mathcal C}=[l_1,\ldots,l_k]$$ ($$l_i>1$$) and satisfy this condition. The determination of $$N_{\mathcal C}(q)$$ amounts to counting the number of roots with pairwise distinct coordinates of a homogeneous quadratic polynomial in $$l_1+\ldots+l_k$$ variables.
Extending studies by C. Wells [J. Comb. Theory 7, 49–55 (1969; Zbl 0165.36701)], the authors prove formulas for $$N_{\mathcal C}(q)$$ with $${\mathcal C}=[4],[2,2],[5]$$. (Already while considering the latter type, the limits of the approach become obvious.) Furthermore, they give a recursive relation for $$N_{[2,2,\ldots,2]}(q)$$ in case of even $$q$$.
In Section 5, the authors discuss their method for arbitrary cycle types. The resulting Proposition 5.1 states that the probability that a permutation of cycle type $$\mathcal C$$ corresponds to a polynomial of degree $$<q-2$$ does not depend on $$\mathcal C$$ asymptotically and estimates it at $$1/q+O(1/q^2)$$. This is contrary to the fact $$N_{[2]}(q)=0$$ for all $$q$$.

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