Power-Free Values of Polynomials

A characterization of almost universal ternary quadratic polynomials with odd prime power conductor

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.

Linearized Polynomials, Galois Groups and Symmetric Power Modules

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.

Density of power-free values of polynomials II

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

We study a function field version of a classical problem concerning square-free values of polynomials evaluated at primes. We show that for a square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density as $n\to \infty$ of primes $P \in \mathbb{F}_q[t]$ of degree $n$ such that $f(P)$ is square-free. Over the integers the analogous result is only known when all irreducible factors of $f$ have degree at most 3.

仅取局部素幂值的多项式

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。

有限域上强大斐波那契多项式

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。

沿多项式幂的格点可见性

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。

The distance to square-free polynomials

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

Let k, r > 1 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t_1...t_r - 1 over the integral points of [1, x]^r.

New identities with Stirling, hyperharmonic, and derangement numbers, Bernoulli and Euler polynomials, powers, and factorials

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.

Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

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.

模素数幂的多项式方程的解

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。

Power Series Approximations to Fekete Polynomials

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.

黎曼 zeta 函数无零点区域的最优余弦多项式

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。

Square-free values of reducible polynomials

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.

在对应于单基因多项式族的某些四次域中计算幂整基

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。

A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial

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 $xy^k+C$

Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$. We also prove a similar result for the $k$-free values of $f(p,q)$ when $p,q\leq H$ are primes. The strongest tool we use is a recent generalization of the determinant method due to Reuss.

Daehee, hyperharmonic, and power sums polynomials

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

A cyclotomic polynomial Φ_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ_{pqr}(x) with p<q<r, p and q fixed and r a free prime.

Power Values of Certain Quadratic Polynomials

In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.

Sums of powers and special polynomials

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

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial f in Z[x] takes infinitely many square-free values. Let c(t) be a polynomial in Fq[t] of degree less than m, and let k < m be co…

Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial

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

For irreducible integer polynomials $f(n)=n^d+c$ we prove an asymptotic formula for the number of $k$-th power free values taken by $f(n)$, for $n$ running up to $x$, subject to the condition $k\ge (5d+3)/9$. This improves earlier results in which the condition was $k\ge (3d+1)/4$. We also show that one can handle $f(p)$ for prime arguments $p$, for the same range of $k$.

Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function

We prove that the Riemann zeta-function $ζ(σ+ it)$ has no zeros in the region $σ\geq 1 - 1/(5.573412 \log|t|)$ for $|t|\geq 2$. This represents the largest known zero-free region within the critical strip for $3.06\cdot10^{10} < |t|<\exp(10151.5)$. Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials.

平均意义下多项式的无平方因子值

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。

Binary polynomial power sums vanishing at roots of unity

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 $f_i$.

Intersective Polynomials Arising from Sums of Powers

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

The purpose of this paper is to generalize this relation of symmetry between the power sum polynomials and the generalized Euler polynomials to the relation between the power sum polynomials and the generalized higher-order Euler polynomials.

Perfect powers in value sets and orbits of polynomials

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.

斐波那契多项式与卢卡斯多项式的若干幂级数推广

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

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

Dwork's congruences for the constant terms of powers of a Laurent polynomial

We prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a p-adic variable satisfies a non-trivial analytic continuation property, similar to what B. Dwork showed for a class of hypergeometric series.

轨道中的幂与多项式生成的半群

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。

Power-free values of polynomials on symmetric varieties

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

We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.

$q$-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping

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…

Generalised polynomials and integer powers

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.