scholarly journals Dickson curves

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Javier Gomez-Calderon
Keyword(s):  
Finite Field ◽  
Monic Polynomial ◽  
Dickson Polynomial

Letkqdenote the finite field of orderqand odd characteristicp. Fora∈kq, letgd(x,a)denote the Dickson polynomial of degreeddefined bygd(x,a)=∑i=0[d/2]d/(d−i)(d−ii)(−a)ixd−2i. Letf(x)denote a monic polynomial with coefficients inkq. Assume thatf2(x)−4is not a perfect square andgcd⁡(p,d)=1. Also assume thatf(x)andg2(f(x),1)are not of the formgd(h(x),c). In this note, we show that the polynomialgd(y,1)−f(x)∈kq[x,y]is absolutely irreducible.

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

2019 ◽  
Vol 18 (04) ◽  
pp. 1950069
Author(s):  
Qian Liu ◽  
Yujuan Sun
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Coding Theory ◽  
Permutation Polynomials ◽  
Combinatorial Designs ◽  
Dickson Polynomial ◽  
Niho Exponents ◽  
Characteristic 3 ◽  
Permutation Trinomials ◽  
Algebraic Forms

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

scholarly journals Chebyshev’s bias in function fields

2008 ◽  
Vol 144 (6) ◽  
pp. 1351-1374 ◽  
Author(s):  
Byungchul Cha
Keyword(s):  
Finite Field ◽  
Number Field ◽  
Central Limit ◽  
Polynomial Ring ◽  
Rational Number ◽  
Function Field ◽  
Sufficient Conditions ◽  
Monic Polynomial ◽  
Field Case ◽  
Necessary And Sufficient

AbstractWe study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In particular, we prove under the GSH that, like the number field case, primes are biased toward quadratic nonresidues. Unlike the number field case, the GSH can be proved to hold in some cases and can be violated in some other cases. Also, under the GSH, we give the necessary and sufficient conditions for which primes are unbiased and describe certain central limit behaviors as the degree of modulus under consideration tends to infinity, all of which have been established in the number field case by Rubinstein and Sarnak.

scholarly journals Polynomial equations for matrices over finite fields

1999 ◽  
Vol 59 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Jiuzhao Hua
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Polynomial Equations

Let E(x) be a monic polynomial over the finite field q of q elements. A formula for the number of n × n matrices θ over q, satisfying E(θ) = 0 is obtained by counting the representations of the algebra q[x]/(E(x)) of degree n. This simplifies a formula of Hodges.

scholarly journals Construction and Determination of Irreducible Polynomials in Galois elds, GF(2m)

2019 ◽  
pp. 1-6
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam ◽  
Fengfan Yang
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Primitive Polynomials

This work is about Construction of Irreducible Polynomials in Finite fields. We defined some terms in the Galois field that led us to the construction of the polynomials in the GF(2m). We discussed the following in the text; irreducible polynomials, monic polynomial, primitive polynomials, eld, Galois eld or nite elds, and the order of a finite field. We found all the polynomials in $$F_2[x]$$ that is, $$P(x) =\sum_{i=1}^m a_ix^i : a_i \in F_2$$ with $$a_m \neq 0$$ for some degree $m$ whichled us to determine the number of irreducible polynomials generally at any degree in $$F_2[x]$$.

scholarly journals Newton slopes for twisted Artin–Schreier–Witt towers

2019 ◽  
Vol 15 (10) ◽  
pp. 2089-2105
Author(s):  
Rufei Ren
Keyword(s):  
Finite Field ◽  
Monic Polynomial ◽  
Arithmetic Progressions ◽  
Character Formula ◽  
Function Form ◽  
Polynomial Formula

We fix a monic polynomial [Formula: see text] over a finite field of characteristic [Formula: see text] of degree relatively prime to [Formula: see text]. Let [Formula: see text] be the Teichmüller lift of [Formula: see text], and let [Formula: see text] be a finite character of [Formula: see text]. The [Formula: see text]-function associated to the polynomial [Formula: see text] and the so-called twisted character [Formula: see text] is denoted by [Formula: see text] (see Definition 1.2). We prove that, when the conductor of the character is large enough, the [Formula: see text]-adic Newton slopes of this [Formula: see text]-function form arithmetic progressions.

Dickson Polynomials Over Finite Fields and Complete Mappings

1987 ◽  
Vol 30 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Gary L. Mullen ◽  
Harald Niederreiter
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Permutation Polynomials ◽  
Dickson Polynomials ◽  
Complete Mapping ◽  
Dickson Polynomial ◽  
Special Cases ◽  
Polynomials Over Finite Fields ◽  
Complete Mappings ◽  
Special Case

AbstractDickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

An O(2n) Example

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Monic Polynomial

This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are made about f: (1) f has 2n distinct roots in k¯, and A₀ = −1; gcd{i¦Aᵢ ≠ = 0} = 1; and (3) f is antipalindromic, i.e., for f pal(x) := x²ⁿf(1/x), we have f pal(x) = −f(x).

Digraphs associated with the kth power map on the quotient ring of polynomials over finite fields

2019 ◽  
Vol 18 (11) ◽  
pp. 1950218
Author(s):  
Amplify Sawkmie ◽  
Madan Mohan Singh
Keyword(s):  
Fixed Point ◽  
Finite Field ◽  
Quotient Ring ◽  
Monic Polynomial ◽  
Directed Edge ◽  
Polynomials Over Finite Fields ◽  
Vertex Set ◽  
Ring Of Polynomials ◽  
Polynomial Formula ◽  
Symmetric Property

Let [Formula: see text] be a monic polynomial over a finite field [Formula: see text], and [Formula: see text] an integer. A digraph [Formula: see text] is one whose vertex set is [Formula: see text] and for which there is a directed edge from a polynomial [Formula: see text] to [Formula: see text] if [Formula: see text] in [Formula: see text]. If [Formula: see text], where the [Formula: see text]’s are distinct monic irreducible polynomials over [Formula: see text], then [Formula: see text] can be factorized as [Formula: see text]. In this work, we investigate the structure of these power digraphs. The semiregularity property is examined, and its relationship with the symmetric property is established. In addition, we look into the uniqueness of factorization of trees attached to a fixed point.

On irreducible polynomials of certain types in finite fields

1969 ◽  
Vol 66 (2) ◽  
pp. 335-344 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Irreducible Polynomials ◽  
Image Position

Let GF(q) be the finite field containing q = pl elements, where p is a prime and l a positive integer. Let P(x) be a monic polynomial in GF[q, x] of degree m. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:(I) P(xr), where r is a positive integer (r-polynomials).(II) xm P(x + x−1). (Reciprocal polynomials.) These have the form(III) xrmP(xr + x−r). (r-reciprocal polynomials.) These have the form Q(xr), where q(x) satisfies (1·1).

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

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