Proof of a Conjecture on Permutation Polynomials over Finite Fields

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained.Comment: 4 page

Further results on complete permutation monomials over finite fields

In this paper, we construct some new classes of complete permutation monomials with exponent $d=\frac{q^n-1}{q-1}$ using AGW criterion (a special case). This proves two recent conjectures in [Wuetal2] and extends some of these recent results to more general $n$'s

A conjecture on permutation trinomials over finite fields of characteristic two

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field \ftwon, a conjecture on permutation trinomials over \ftwon[x] proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$

Some new results on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int. J. Inf. Coding Theory, 2010, 1, pp. 149-170).Comment: 21 pages. We have changed the title of our pape

Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field

On the Bateman-Horn conjecture for polynomials over large finite fields

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$) polynomials $F_1,\ldots,F_m\in\mathbf{F}_q[t][x]$, with $q$ odd, we show that the number of $f\in\mathbf{F}_q[t]$ of degree $n\ge\max(3,\mathrm{deg}_t F_1,\ldots,\mathrm{deg}_t F_m)$ such that all $F_i(t,f)\in\mathbf{F}_q[t],1\le i\le m$ are irreducible is $$\left(\prod_{i=1}^m\frac{\mu_i}{N_i}\right) q^{n+1}\left(1+O_{m,\,\max\mathrm{deg} F_i,\,n}\left(q^{-1/2}\right)\right),$$ where $N_i=n\mathrm{deg}_xF_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$ and $\mu_i$ is the number of factors into which $F_i$ splits over $\bar{\mathbf{F}_q}$. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over $\mathbf{F}_q(t)$) polynomials $F_1,\ldots,F_m$ not necessarily monic in $x$ under the assumptions that $n$ is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve $C$ defined by the equation $$\prod_{i=1}^mF_i(t,x)=0$$ (this number is always bounded above by $\left(\textstyle\sum_{i=1}^m\mathrm{deg} F_i\right)^2/2$, where $\mathrm{deg}$ denotes the total degree in $t,x$) and $$p=\mathrm{char}\,\mathbf{F}_q>\max_{1\le i\le m} N_i,$$ where $N_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$

More Classes of Complete Permutation Polynomials over \F_q

In this paper, by using a powerful criterion for permutation polynomials given by Zieve, we give several classes of complete permutation monomials over \F_{q^r}. In addition, we present a class of complete permutation multinomials, which is a generalization of recent work.Comment: 17 page

The Weil bound and non-exceptional permutation polynomials over finite fields

A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace $n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we further refine the upper bound for $q$ with respect to $n$, by a more concise and direct proof following Wan's arguments.Comment: 5 page

A Class of Binomial Permutation Polynomials

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

Linearized polynomial maps over finite fields

We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible linearized polynomial maps correspond to invertible matrices. This alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures (most notably, the Jacobian Conjecture) for this kind of polynomials and polynomial maps.Comment: 21 pages; added references, object name in title more in line with literature, modified setup to put results in clearer perspective, but no change in result