Olivier Rahavandrainy
We give all bi-unitary non splitting even perfect polynomials over the prime field of two elements, which are divisible by Mersenne irreducible polynomials raised to special exponents. We also identify all bi-unitary perfect polynomials over the same field, with at most four irreducible factors. We then complete, in this manner, a list given by J.T. B. Beard Jr.
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Unitary super perfect numbers
我们将证明9，165都是奇酉超完满数。
All unitary perfect polynomials over $F_2$ with less than five distinct prime factors
我们找到了素域$\F_2$上所有具有小于5个不同素因子的酉完全多项式。
We shall give some results for an integer divisible by its unitary totient.
The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.
Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors
本文确定了有限域 $\mathbb{F}_4$ 上所有至多包含四个不可约因子的非分裂双酉完美多项式，并指出此类因子有无限多个。
On splitting bi-unitary perfect polynomials over $\mathbb{F}_{p^2}$
本文研究了有限域上分裂双酉完全多项式的分类，给出了 $\mathbb{F}_4$ 上的全部解和奇素数 $p$ 时 $\mathbb{F}_{p^2}$ 上的部分解。
All bi-unitary perfect polynomials over $\F_2$ with at most four irreducible factors
该研究确定了有限域$\text{F}_2$上至多含四个不可约因子的双酉完全多项式的全部类型
All bi-unitary perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors
该研究针对二元素域$\boldsymbol{\text{F}}_2$，找到了仅以梅森多项式为奇不可约因子的所有非分裂双酉完美多项式，是经典算术问题的多项式变体研究成果。
We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $σ(A)/A $ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$.
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution.
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $Φ_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m\ge 1$ we show that every integer occurs as a coefficient of $Φ^*_{mn}(x)$ for some $n\ge 1$. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/Φ_n^*(x),$ the inverse unitary cyclotomic polyno…
A unitary cyclotomic polynomial of order three is a polynomial of the form \[ Φ^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$, $q$ and $r$. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of $P=p^a$, $Q=q^b$ and $R=r^c$. We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of $\mathbb{Z}$.
Even perfect polynomials over $F_2$ with four prime factors
二进制域$\F_2$上的完美多项式是等于其所有因子之和的多项式$A \in \F_2[x]$。如果是$\gcd(A,x^2-x) \neq 1$，则称$A$为偶数。已知$\F_2$上最多有3个素因子的所有偶完全多项式的列表。本文的目的是给出$\F_2$上所有具有四个素因子的偶完全多项式的列表。这些是所有已知的在$\F_2$上具有四个素因子的完美多项式。
On special perfect polynomials over $\mathbb{F}_2$
本文给出了 $\mathbb{F}_2$ 上特殊完全多项式满足的一些必要条件。
The paper is about an arithmetic problem in $\F_2[x]$. We give \emph{admissible} (necessary) conditions satisfied by a set of odd prime divisors of perfect polynomials over $\F_2$. This allows us to prove a new characterization of \emph{all} known perfect polynomials, and to open a way of finding more of them (if they exist).
Search techniques for root-unitary polynomials
我们给出了一个轶事讨论的问题，寻找多项式的所有根上的单位圆，其系数是有理数受一定的同余条件。我们说明了一个例子，从计算中的p-adic上同调由雅培，Kedlaya，和罗伊，在其中（使用与安德烈Wibisono开发的实现），我们恢复了一个表面的zeta函数在有限域。
Perfect cuboids and multisymmetric polynomials
一个完美的欧拉长方体是一个长方体，它有整数条边和整数条面对角线，其空间对角线也是整数。寻找这种平行六面体或证明其不存在的问题是一个古老的未解决的数学问题。理想欧拉长方体的丢番图方程具有显式的$S_3$对称性。本文将长方体方程关于其$S_3$对称性用多对称多项式进行因式分解。一些因子方程显式计算。
Perfect cuboids and irreducible polynomials
构造完美长方体的问题与某类具有三个整数参数$a$、$b$和$u$的一元多项式有关。它们在整数环上对$a$、$b$和$u$的某些限制下的不可约性意味着不存在完美长方体。这种不可约化被证明，然后对大约10000个$a$、$b$和$u$的实例进行数值验证。
Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a quartic polynomial for which the product of it's first consecutive values is infinitely often a perfect square.
On splitting perfect polynomials over $\mathbb{F}_{p^2}$
研究了$\mathbb{F}_{p^2}$上分裂完全多项式中各项指数的性质，其中$p$是素数.这推广了Beard等人在$\mathbb{F}_p$上的工作。更正的纸。旧的引理2.17到2.20在出版的版本中需要在这里做一个修正。
We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the $abc$-Conjecture for number fields, we obtain a finiteness result for powers in ratios of arbitrary elements in orbits.
On an ideal of multisymmetric polynomials associated with perfect cuboids
一个完美的欧拉长方体是一个长方体，它有整数边，整数面对角线，整数空间对角线。找到这样的平行六面体或证明它们不存在是一个古老的未解决的数学问题。从代数上讲，这个问题可以用一组丢番图方程来描述。长方体问题的对称性方法是基于长方体问题的丢番图方程的自然$S_3$对称性。将这些方程关于它们的$S_3$对称性进行因式分解，得到多对称多项式环内的某些理想。在本文中，这个理想是完全计算和提出通过其基础。
On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices
证明了随机酉矩阵的独立特征多项式的任意有限线性组合的零点几乎都在单位圆上。这一结果是随机矩阵模拟的早期结果由Bjueri和Hejhal的线性组合的$L$ -功能的零的分布，从而提供了进一步的证据证明之间的关系的特征多项式的值分布的随机酉矩阵和值分布的$L$ -功能的临界线。
The bimodal distribution in the derivative of unitary polynomials
本文研究单位圆上多项式导数零点的双峰分布现象，识别出第二峰的可能来源，并通过随机酉矩阵特征多项式的数值例子加以支持。
Uniform asymptotics of the coefficients of unitary moment polynomials
Keating和Snaith证明了单位圆上随机酉矩阵的特征多项式的$2k^{th}$绝对矩由$k^2$次多项式给出。在这篇文章中，该多项式的系数的一致渐近推导，并找到一个最大的系数。某些渐近性以显式形式给出。数值数据来支持这些计算。随机矩阵理论和黎曼zeta函数之间的一些明显的联系进行了讨论。
Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.
Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble
本文基于对称函数理论和 Borodin-Olshanski-Strahov 公式，推导出 ACUE 特征多项式高阶混合矩与比值平均的闭式公式，并与 CUE 的结果进行了比较。
Colored base-3 partitions, sequences of polynomials, and perfect numbers
受含偶完全数子序列的三进制着色分拆计数启发，研究者定义四变量多项式序列并研究其性质，探讨两类切比雪夫多项式相关子序列，还分析了具组合意义与特殊零点分布的单变量多项式序列。
We provide a combinatorial derivation of an asymptotic formula for averages of mixed ratios of characteristic polynomials over the unitary group, where mixed ratios are products of ratios and/or logarithmic derivatives. Our proof of this formula is a generalization of Bump and Gamburd's elegant combinatorial proof of Conrey, Forrester and Snaith's formula for averages of ratios of characteristic polynomials over the unitary group. One application of this formula is an asymptotic expression for sums over zeros of a random characteristic polynomial from the unitary group, which we call an explic…
Twisted moments of characteristic polynomials of random matrices in the unitary group
本文提出酉群随机矩阵特征多项式扭曲矩的严格理论，证明相关 Schur 多项式恒等式，为黎曼 zeta 函数矩的启发式方法提供概念验证。
On the logarithmic derivative of characteristic polynomials for random unitary matrices
本文研究 Haar 随机酉矩阵特征多项式的对数导数，证明其在单位圆附近可由邻近零点逼近，并在远离单位圆处建立介观中心极限定理，分别类比 Riemann zeta 函数的相关结果。
Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups
本文证明了完美李群商中某些单位元子群轨道的有效等分布定理，并应用该结果得到不定二次型的有效Oppenheim猜想版本。
In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.
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