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Register a new userA Class of Binomial Permutation Polynomials
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Added by
Chunlei Li
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.
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The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line $Re,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Goulds S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their `critical polynomial factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend these results to a $1$-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;beta)=pm p_n(1-s;beta)$, similar to that for the Riemann xi function. It is shown that via manipulation of the binomial factors, these `critical polynomials can be simplified to an S:3/2 form, which after normalisation yields the rational function $q_n(s).$ The denominator of the rational form has singularities on the negative real axis, and so $q_n(s)$ has the same `critical zeros as the `critical polynomial $p_n(s)$. Moreover as $srightarrow infty$ along the positive real axis, $q_n(s)rightarrow 1$ from below, mimicking $1/zeta(s)$ on the positive real line. In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial form, and with $mathcal{C}_n$ the $n$th Catalan number, $s$ an integer, we show that polynomials $4mathcal{C}_{n-1}p_{2n}(s)$ and $mathcal{C}_{n}p_{2n+1}(s)$ yield integers with only odd prime factors. The results touch on analytic number theory, special function theory, and combinatorics.
Let $f(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})inBbb F_{q^2}[X]$, where $a,binBbb F_{q^2}^*$. In a series of recent papers by several authors, sufficient conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial (PP) of $Bbb F_{q^2}$ and, in characteristic $2$, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when $text{char},Bbb F_q=3$, $f$ is a PP of $Bbb F_{q^2}$ if and only if $(ab)^q=a(b^{q+1}-a^{q+1})$ and $1-(b/a)^{q+1}$ is a square in $Bbb F_q^*$.
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{frac{q^2 -1}{3}+1} +x$ over $mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{frac{q^2 -1}{d}+1} -bx$ over $mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form [ f(x)=(ax^{q} +bx +c)^r phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~text{over $mathbb{F}_{q^2}$}, ] where $a,b,c,u,v in mathbb{F}_{q^2}$, $r in mathbb{Z}^{+}$, $phi(x)in mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.
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