Kostadinka Lapkova, Stanley Yao Xiao
We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$ for any $n\geq 1$, including when the variables are prime, as long as $k\geq (3d+1)/4$. Thus we generalize a work of Browning, while we use a different sieveing technique for the middle range of primes.
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It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $Φ_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show t…
Square-free values of random polynomials
本文在平均意义上研究了任意度数整系数多项式是否取无穷多个无平方因子值的问题，突破了以往仅无条件解决度数不超过3多项式的限制。
Counting square-free values of random polynomials
本文研究了随机多项式平方自由值的计数问题，证明了平均误差项与主项的四次方根关系。
We investigate the density of square-free values of polynomials with large coefficients over the rational function field $\mathbb{F}_q[t]$. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial $N$ as a sum of a small $k$-th power and a square-free polynomial.
Building upon the work of A. Booker and C. Pomerance (2017), we prove that for a prime power $q \geq 7$, every residue class modulo an irreducible polynomial $F \in \mathbb{F}_q[X]$ has a non-constant, square-free representative which has no irreducible factors of degree exceeding $\text{deg}~F -1$. We also give applications to generating sequences of irreducible polynomials.
Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.
Power-Free Values of Polynomials
设f是d ≥ 3次的不可约多项式，没有固定的素除子.我们得到了使得f（p）是（d-1）-free的素数p<x的个数的一个渐近公式。
A characterization of almost universal ternary quadratic polynomials with odd prime power conductor
一个整二次多项式（具有正定二次部分）称为几乎泛多项式，如果它表示除1/2个正整数以外的所有正整数。本文引入了二次多项式的导体，并给出了几乎泛三元二次多项式具有奇素数幂导体的一个有效刻画。
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.
In this paper we prove that polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ of degree $d \geq 3$, satisfying certain hypotheses, take on the expected density of $(d-1)$-free values. This extends the authors' earlier result where a different method implied the similar statement for polynomials of degree $d\geq 5$.
Square-free values of polynomials evaluated at primes over a function field
我们研究了一个经典问题的函数域版本，该问题涉及多项式在素数处的无平方值。我们证明了对于无平方多项式$f\in \mathbb{F}_q[t][x]$，存在一个极限密度为素数$P \in \mathbb{F}_q[t]$的次数$n$的$n\to \infty$，使得$f(P)$是无平方的.在整数上，只有当$f$的所有不可约因子的次数最多为3时，才知道类似的结果。
Polynomials assuming only local prime powers
针对将局部域映射到其p次幂子集的多项式，研究给出该类多项式的刻画，证明其范围远大于多项式的p次幂类。
Powerful Fibonacci polynomials over finite fields
本文研究了有限域上斐波那契多项式及广义卢卡斯多项式成为完全幂或强大多项式的特征。
Lattice point visibility along powers of polynomials
研究多项式视线下的格点可见性，证明了Chaubey和Pandey的可见性密度猜想，适用于广泛多项式类。
In this paper, we consider a variant of Turán's problem on the distance from an integer polynomial in $\mathbb{Z}[x]$ to the nea\-rest irreducible polynomial in $\mathbb{Z}[x]$. We prove that for any polynomial $f \in \mathbb{Z}[x]$, there exist infinitely many square-free polynomials $g\in \mathbb{Z}[x]$ such that $L(f-g) \le 2$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \le 1$. For this, for each integer $d \geq 16$ we construct infinitely many polynomials $f \in \mathbb{Z}…
Power-free values of the polynomial t_1...t_r - 1
设k，r > 1是两个整数。我们证明了一个渐近公式的数量k-自由值的r变量多项式t_1…t_r - 1在[1，x]^r的整数点上的分布。
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials.
For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of $f$. We also describe zero-free lens-shaped regions for $f$ that are associated to the zero-free sectors of the reciprocal of $f$. As an application, we use these zero-free regions to obtain irreducibility criteria for polynomials with integer coefficients that take a prime or a prime power value.
Solutions of a polynomial equation modulo a prime power
本文探讨了如何求解多项式方程模整数（特别是素数幂）的整数解问题。
We study how well Fekete polynomials $$ F_p(X) = \sum_{n=0}^{p-1} \left(\frac{n}{p}\right) X^n \in {\mathbb Z}[X] $$ with the coefficients given by Legendre symbols modulo a prime $p$, can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.
Optimal Cosine Polynomials for Riemann Zeta Zero-Free Region
本文用数值方法确定次数为 7 和 8 的最优余弦多项式，并给出黎曼 zeta 函数相应的无零点区域，推进了 Landau 构造下最大无零点区域问题的研究。
We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.
Calculating power integral bases in some quartic fields corresponding to monogenic families of polynomials
本研究针对J. Harrington、L. Jones及P. Voutier提出的单基因四次多项式族，刻画其生成数域中幂整基的全部生成元，补充了相关单基因数域的幂整基生成元研究。
In this paper, a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools are Watson's $_8ϕ_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials. By taking suitable parameter substitutions in the established $q$-supercongruence, some nice congruences involving the Bernoulli numbers are derived.
On the $k$-free values of the polynomial $xy^k+C$
考虑$k\geq 2$的多项式$f(x,y)=xy^k+C$和任何非零整数常数$C$。当$x, y\leq H$满足时，我们导出了$f(x,y)$的无$k$值的渐近公式.当$p,q\leq H$是素数时，我们也证明了$f(p,q)$的$k$ -free值的类似结果.我们使用的最强有力的工具是最近推广的行列式方法由于罗伊斯。
In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial $S_k(x)$ associated with the sum of $k$th powers of the first $n$ positive integers $S_k(n) = 1^k + 2^k + \cdots + n^k$. Furthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic…
The family of ternary cyclotomic polynomials with one free prime
一个分圆多项式Φ n（x）称为三元多项式，如果n=pqr，p，q，r是不同的奇素数.三元分圆多项式是最简单的多项式，其系数的行为尚未完全理解。本文建立了多项式族Φ_{pqr}（x）中系数的一些结果，并给出了一些公式，其中p<q<r，p和q是固定的，r是自由素数。
Power Values of Certain Quadratic Polynomials
本文计算了具有负无平方判别式的二次多项式$f$的$q$次幂值，使得$q$与$\mathbb{Q}$上$f$的分裂域的类数互质.唯一的因式分解和整数序列的原始因子的理论是用来推导出一个界的值$q$这是足够小，以允许其余的情况下，很容易检查。结果被用来确定某些多项式生成的整数序列，包括西尔维斯特序列的所有完美的权力条款。
In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and also show their connection to other interesting series. In particular, we show their connection to an interesting problem of Ovidiu Furdui.
Square-free values of multivariate polynomials over function fields in linear sparse sets
设f是Fq[t][x]中的无平方多项式，其中Fq是q个元素的域。我们将f视为变量x的多项式，其系数在环Fq[t]中。本文研究了由线性条件给出的Fq[t]的稀疏子集中f的无平方值。我们研究的动机是一个用整数多项式表示无平方数整数的类似问题，其中证明了，抛开一些简单的例外情况，Z[x]中的无平方数多项式f取无穷多个无平方数值。设c（t）是Fq[t]中次数小于m的多项式，且k我们的主要结果的一个后果，是，如果q是足够大的关于m和度的f在t和x，则存在$β_1,β_2$在Fq，使$f(t,c(t)+β_1t^k+β_2)$是平方自由。此外，当q趋于无穷大时，最后一个对于Fq中几乎所有的$β_1$和$β_2$都成立。主要结果表明，类似的结果也适用于其他情况。然后将结果推广到多元多项式。
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Φ_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
Powerfree Values of Polynomials
对于不可约的整数多项式$f(n)=n^d+c$，我们证明了一个渐近公式的数目$k$次幂自由值所采取的$f(n)$，$n$运行到$x$，受条件$k\ge (5d+3)/9$。这改进了早期的结果，其中条件是$k\ge (3d+1)/4$。我们还表明，一个可以处理$f(p)$的素数参数$p$，对于相同的范围的$k$。
Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function
证明了黎曼ζ函数$ζ(σ+ it)$在$|t|\geq 2$的区域$σ\geq 1 - 1/(5.573412 \log|t|)$中没有零点.这代表了$3.06\cdot10^{10} < |t|<\exp(10151.5)$临界条带内最大的已知无零区域。我们的改进是通过确定一些具有特定性质的有利三角多项式以及分析卡迪里方法中的误差项而得到的。我们还改进了Landau关于非负三角多项式问题的一个上界。
Square-free values of polynomials on average
本文针对多项式无平方因子值计数的平均猜想，改进了取值长度x与多项式高度的依赖关系，方法是调整已有研究中提出的相关猜想研究方法。
Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$. We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.
Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every positive integer.
Symmetry properties of the generalized higher-order Euler polynomials
本文的目的是将幂和多项式与广义欧拉多项式之间的对称关系推广到幂和多项式与广义高阶欧拉多项式之间的对称关系。
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.
Some power series extensions of Fibonacci and Lucas polynomials
本研究聚焦形式幂级数领域，探讨可作为偶/奇下标斐波那契、卢卡斯多项式插值的形式幂级数，拓展了相关多项式的幂级数研究方向。
Factoring polynomials in the ring of formal power series over Z
本文研究了整系数多项式在Z上形式幂级数环Z[[x]]中的因式分解性质。本文讨论了任意次数的多项式，并给出了其约化为幂级数的充分条件。此外，如果多项式在Z[[x]]上可约，我们提供了一个显式因式分解算法。对于常数项是素数幂的多项式，我们的研究引出了p进整数的讨论。
Dwork's congruences for the constant terms of powers of a Laurent polynomial
证明了Laurent多项式的幂的常数项满足模素幂的某些同余式。作为推论，这些数的生成级数被认为是p-adic变量的函数，满足非平凡的解析延拓性质，类似于B。一类超几何级数的Dwork证明。
Powers in orbits and semigroups generated by polynomials
本文研究数域上多项式轨道及半群中完全幂的存在性，修正并推广了Ahmadi等人的结果，提出有限算法检测乘法平移，并给出轨道元素的有限性结论。
Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.
Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power
通过考察包含$f(X)$的覆盖环${\rm Int}(\mathbb{Z})$的极大理想的幂的收缩，刻画了$\mathbb{Z}[X]$中多项式$f(X)$的固定因子.给定一个素数$p$和一个正整数$n$，我们还得到了$\mathbb{Z}[X]$中固定因子可被$p^n$整除的多项式的理想的一个完全刻画.
By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'Hôpital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercon…
We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) = 0$ for all $n \geq 0$ then there exists infinitely many $m \in \mathbb{N}$, not divisible by $k$, such that $g(mk^n) = 0$ for some $n \geq 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.
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