In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.
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.
We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.
We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well…
In this note we give the Kolyvagin recursion in cyclotomic Euler systens a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of the chosen field satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution a la Kubert. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohom…
Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then $f(a^n)=0$. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases $K=\mathbb{F}_p$ and $K=\mathbb{Q}$.
By using the piecewise method, Lagrange interpolation formula and Lucas' theorem, we determine explicit expressions of the inverses of a class of reversed Dickson permutation polynomials and some classes of generalized cyclotomic mapping permutation polynomials over finite fields of characteristic three.
This paper examines the linear complexity of new generalized cyclotomic binary sequences of period $2p^n$ recently proposed by Yi Ouang et al. (arXiv:1808.08019v1 [cs.IT] 24 Aug 2018). We generalize results obtained by them and discuss author's conjecture of this paper.
Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm Φ_{p_1}(\pm x)Φ_{p_2}(\pm x^{p_1})\cdots Φ_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots p_{r}$ and the $p_i$ are primes, not necessarily distinct. Here $Φ_p(x):=(x^p-1)/(x-1)$ is the $p-$th cyclotomic polynomial. In \cite{1}, they also proved the conjecture for $N$ odd or a power of 2. In this paper we introduce a so-called $E-$transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to inve…
In this (mostly historical) note we show how a unified Kummer-Artin-Schreier sequence from [W], [SOS] may be recovered from the relativistic velocity addition law.
We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new consequences, including a necessary condition for the algebraic solution by radicals of certain irreducible polynomials.
In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(ζ\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.
The $k$-higher Mahler measure of a nonzero polynomial $P$ is the integral of $\log^k|P|$ on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for $k=1$) for $k>1$ and find some interesting formulae for 2- and 3-higher Mahler measure of cyclotomic polynomials.
This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m,n$, the conjecture is that either $Φ_m(q)\leqslantΦ_n(q)$ or $Φ_m(q)\geqslantΦ_n(q)$ holds for all integers $q\geqslant 2$. Pomerance and Rubinstein-Salzedo proved the conjecture in a preprint called `Cyclotomic coincidences' [arXiv:1903.01962].
In this paper, we study the fine Selmer groups attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic p. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic $\Zp$-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups.
We present an elementary identity for the cyclotomic polynomials $Φ_n(X)$ which reflects a kind of multiplicative property of $Φ_n(X)$ as a function of $n$, and we explore its connections with the properties of other arithmetical functions. Important Note: In the first version of this article uploaded to the arXiv, it is said that this result seemed to be new. However, after that, I have learned that this identity (in Theorem 1.1) has previously appeared in Cheng, C. C. A., McKay, J. H., & Wang, S. S. S. (1995). Resultants of cyclotomic polynomials. Proceedings of the American Mathematical Soc…
Let $q=4$ and $k$ a positive integer. In this short note, we present a class of permutation polynomials over $\Bbb F_{q^{3k}}$. We also present a generalization.
Let $Φ_n(X)$ and $Ψ_n(X)=\frac{X^{n}-1}{Φ_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$ such that $ 0 \leq l_{1} \leq φ(d_{1})-1 $ and $ 0 \leq l_{2} \leq φ(d_{2})-1 $, we show that \[\left \langle X^{l_{1}} Ψ_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} Ψ_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $ \langle \cdot, \cdot \rangle $ is the inner product on $\mathbb{Q}[X]$ defined by $ \langle \sum_{k} a_{k}X^{k},\sum_{k} b_{k}X…
Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$
In this note, we describe a conjecture, that, for an odd prime p, relates special values of a cup product pairing on cyclotomic p-units in the pth cyclotomic field to the L-values of newforms satisfying modulo p congruences with Eisenstein series.
We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$ that fix $0$ and restrict to the multiplication $x\mapsto a_ix$ by a fixed element $a_i\in\mathbb{F}_q$ on each coset $C_i$ of a given subgroup $C$ of $\mathbb{F}_q^{\ast}$. The gist of two of our main results is that as long as $q$ is large enough relative to the index $|\mathbb{F}_q^{\ast}:C|$, all cycle types of first-order cyclotomic permutations with only …
Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $Φ_n$ denote the $n$-th cyclotomic polynomial and let $Ψ_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(Φ_n)$ and $g(Ψ_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(Φ_{p_1})$, $g(Ψ_{p_1})$ and $g(Ψ_{p_1p_2})$. Hence the simplest non-trivial cases are $g(Φ_{p_1p_2})$ and $g(Ψ_{p_1p_2p_3})$. We provide an exact expression for $g(Φ_{p_1p_2}).$ We also provide an exact expression for $g(Ψ_{p…
In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves o…
Let $p$ be an odd prime and $e$ be a positive integer. We completely explain the permutation binomials and trinomials arising from the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over $\mathbb{F}_{p^e}$ when $n=p^l+2$, where $l\in \mathbb{N}$.
Permutation polynomials over finite fields have taken an important role in vast areas in mathematics as well as engineering. Recently, Tu et al. gave some classes of complete permutation polynomials over finite fields of even characteristic. In this paper, we give some properties based on a work of Akbary et al. With the help of the properties, we give extensions of Tu et el.'s results to arbitrary characteristic.
Let $F$ be a number field. Let $p$ be a prime number. Washington proved the $\ell$-part of the class numbers in cyclotomic $\mathbb{Z}_p$ extension of $F$ is bounded when $F$ is an abelian number field and $\ell\neq p$ is a prime. By class number formula, this is essentially a mod $\ell$ nonvanishing property of Drichlet L-functions at $s=0$. In \cite{Sinnot}, Sinnot gave a different proof by algebraic methods. In this article, we show that Sinnot's method can prove nonvanishing properties of Drichlet L-functions at $s=-k$, where $k\geq 0$ is an integer. Since the Lichtenbaurn conjecture which…
The Euler-Mascheroni constant $γ=0.5772\dots\!$ is the $K=\mathbb{Q}$ example of an Euler-Kronecker constant $γ_K$ of a number field $K.$ In this note we consider the size of the $γ_q=γ_{K_q}$ for cyclotomic fields $K_q:=\mathbb{Q}(ζ_q).$ Assuming the Elliott-Halberstam Conjecture (EH), we prove uniformly in $Q$ that $$\frac{1}{Q}\sum_{Q<q\le 2Q} \left |γ_q - \log q\right |= o(\log Q).$$ In other words, under EH the $γ_q / \log q$ in these ranges converge to the one point distribution at $1$. This theorem refines and extends a previous result of Ford, Luca, and Moree for prime $q.$
Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these polynomials in the case that every prime divisor of $n$ divides $q-1$. This result generalizes the results find in Chou, W.S., The Factorization of Dickson polynomials over finite fields. Finite Fields Appl. {\bf3} (1997) 84-96, Fitzgerald R. W., Yucas J. L., Explicit factorization of cyclotomic and Dickson polynomials over finite fields. Arithmetic of 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.
We show that all of the "new" permutation polynomials in the recent paper arXiv:2207.13335 (H. Song et al.) are in fact known. We also present a new type of question in this area.
We give historical remarks related to arXiv:2112.14547 ("A New Method of Construction of Permutation Trinomials with Coefficients 1", by Guo et al.). In particular, we show that the "new" permutation polynomials in that paper are actually well known. In addition we give a simpler derivation of these permutation polynomials than had been given previously, which demonstrates the general method of producing permutation polynomials that was introduced in arXiv:1310.0776.
We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$, the maximal cross-correlation of the family of the $φ(p-1)$ different Welch Costas permutations of $\{1,\ldots,p-1\}$ is $(p-1)/t$, where $t$ is the smallest prime divisor of $(p-1)/2$ if $p$ is not a safe prime and at most $1+p^{1/2}$ otherwise. Here $φ$ denotes Euler's totient function and a prime $p$ is a safe prime if $(p-1)/2$ is also pr…
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.
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.
Define a permutation $σ$ to be coprime if $\gcd(m,σ(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c = \prod_{p\text{ prime }}\frac{(p-1)^{2(1-1/p)}}{p\cdot (p-2)^{(1-2/p)}}.\] The techniques involve entropy maximization for the upper bound, and a mixture of number-theoretic bounds, permanent estimates, and the absorbing method for the lower bound.
For q > 2, Carlitz proved that the group of permutation polynomials (PPs) over F_q is generated by linear polynomials and x^{q-2}. Based on this result, this note points out a simple method for representing all PPs with full cycle over the prime field F_p, where p is an odd prime. We use the isomorphism between the symmetric group S_p of p elements and the group of PPs over F_p, and the well-known fact that permutations in S_p have the same cycle structure if and only if they are conjugate.
This paper mainly studies problems about so called "permutation polynomials modulo $m$", polynomials with integer coefficients that can induce bijections over Z_m={0,...,m-1}. The necessary and sufficient conditions of permutation polynomials are given, and the number of all permutation polynomials of given degree and the number induced bijections are estimated. A method is proposed to determine all equivalent polynomials from the induced polynomial function, which can be used to determine all equivalent polynomials that induce a given bijection. A few problems have not been solved yet in this…
