In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $μ_j,$ then $A$ is irreducible with probability tending to $1$ as $n\to\infty$ under a condition of near-uniformity of the $μ_j$ modulo four primes (which notably happens when the $μ_j$ are uniform over a segment of $\mathbb Z$ of length $N\ge 35.$) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound $\mathbb{P}(A \text{ irreducible}) = 1 -O(1/\sqrt n)$ when we have near-uniformity modulo twelve primes, which notably happens when the $μ_j$ are uniform over a segment of length $N\ge 10^8.$
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.
We study certain kind of polynomials associated with Lissajous curves, called Chebyshev-Lissajous polynomials. We investigate their irreducibilities over the real numbers and complex numbers, thus comfirming two conjectures proposed by Merino.
In this paper, we show that for each $n\geq 1$, the generalised Hermite-Laguerre Polynomials $G_{\frac{1}{4}}$ and $G_{\frac{3}{4}}$ are either irreducible or linear polynomial times an irreducible polynomial of degree $n-1$.
The Krawtchouck polynomials arise naturally in both coding theory and probability theory and have been studied extensively from these points of view. However, very little is known about their irreducibility and Galois properties. Just like many classical families of orthogonal polynomials (e.g. the Legendre and Laguerre), the Krawtchouck polynomials can be viewed as special cases of Jacobi polynomials. In this paper we determine the Newton Polygons of certain Krawtchouck polynomials and show that they are very similar to those of the Legendre polynomials (and exhibit new cases of irreducibilit…
We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular, we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is either an Eneström-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.
We consider the irreducibility of Generalised Laguerre Polynomials for negative integral values given by $L_n^{(-1-n-s)}(x)=\displaystyle\sum_{j=0}^{n}\binom{n-j+s}{n-j}\frac{x^j}{j!}.$ For different values of $s,$ this family gives polynomials which are of great interest. It was proved earlier that for $0 \leq s \leq 60,$ these polynomials are irreducible over $\mathbb{Q}.$ In this paper we improve this result upto $s \leq 88.$
We give a new proof of the Brawley-Carlitz theorem on irreducibility of the composed products of irreducible polynomials. Our proof shows that associativity of the binary operation for the composed product is not necessary. We then investigate binary operations defined by polynomial functions, and give a sufficient condition in terms of degrees for the requirement in the Brawley-Carlitz theorem.
In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever reducible. As a consequence, we give a simple procedure for checking the irreducibility of trinomials of this form and separability criterion for certain quadrinomials.
Let $E_k$ be the normalized Eisenstein series of weight $k$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Let $φ_k$ be the polynomial that encodes the $j$-invariants of non-elliptic zeros of $E_k$. In 2001, Gekeler observed that the polynomials $φ_k$ seem to be irreducible (and verified this claim for $k\leq 446$). We show that $φ_k$ is irreducible for infinitely many $k$.
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.
A second order polynomial sequence is of Fibonacci-type $\mathcal{F}_{n}$ (Lucas-type $\mathcal{L}_{n}$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are irreducible if and only if $n$ is a prime number. For example, the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials are irreducible when $n$ is a prime number; and Chebyshev polynomials (second kind), Morgan-Voyce polynomials (Fibonacci type), and Vieta polynomials are reducible when…
Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k for the full modular group. We show that if there exists n>1 such that T_{n,k}(X) is irreducible and has the full symmetric group as Galois group, then the same is true of T_{p,k}(X) for all primes p.
We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$, then the proportion of $n$-tuples $(t_1,\ldots, t_n)$ of monic polynomials of degree $d$ for which $f(t_1,\ldots, t_n,X)$ is reducible out of all $n$-tuples of degree $d$ monic polynomials is $O(dq^{-d/2})$.
Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has the same Galois group as $P$. We discuss here a method for obtaining an explicit description of the set of exceptional numbers $t$, i.e., those for which $P(t,X)$ is either reducible or has a different Galois group than $P$. To illustrate the method we determine the exceptional specializations of two polynomials of degrees four and six.
For a rational $q=u+\fracα{d}$ with $u, α, d\in \ACOBZ$ with $u\ge 0, 1\le α<d$, $\gcd(α, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(α+(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(α+(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(α+(i+u)d)\right) \end{align*} where $a_0, a_1, \cdots, a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also prove a…
For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes $p$ such that all iterations of $f$ are irreducible modulo $p$ is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval $[1, Q]$ as $Q \to \infty$. For this class of polynomials this gives a more precise version of a recent result of A. Ferraguti (2018), which applies t…
We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $\mathbb Q$, where $c, d \in \mathbb Z[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N \ge 5$ and $k \in \{3, 4, \ldots, 24\} \setminus \{9, 16\}$. We also give a complete description of the factorization of polynomials of the form $x^N + …
For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of $P_{n,k}(x)$ over the field of rational numbers when $2\leqslant 2k \leqslant n<(k+1)^3$.
The famous irreducibility criteria of Schönemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of Schönemann-Eisenstein-Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.
For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $α=u$ with $1\leq u \leq 50$ or $α=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $ψ_n^{(α)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},α=89/2$ are necessary. The above result with $0\leqα\leq 10$ is due to Filaseta, Finch and Leidy and with $α\in \{-1/2,1/2\}$ due to Schur.
For a positive integer $n$ and a real number $α$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(α)}_n(x)=\sum^n_{j=0}\frac{(n+α)(n-1+α)\cdots (j+1+α)(-x)^j}{j!(n-j)!}. \end{align*} These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of $L^{(\pm \frac{1}{2})}_n(x)$ and $L^{(\pm \frac{1}{2})}_n(x^2)$ and derived that the Hermite polynomials $H_{2n}(x)…
In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over $\mathbb{Q}$. In particular, we obtain infinitely many new examples of the phenomenon studied in \cite{Jones}. While doing this, we also find, up to three possible exceptions, all integers $a$ such that all iterates of the quadratic polynomial $(x+a)^2-a-1$ are irreducible over $\mathbb{Q}$, which answers a question posed in \cite{AyadMcdonald}, except for three v…
We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform (DFT) of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the DFT of characteristic elementary symmetric functions (which produce the coefficients of characteristic polyn…
In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb{Q}(T_1, \ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$ specialisations $\mathbf{t} \in \mathbb{Z}^s$ with $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.
We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square. Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials. We also explain how irreducibili…
We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is $L_{n}^{(-1-n-r)}(x)= \sum\limits_{j=0}^{n} \binom{n-j+r}{n-j} \frac{x^{j}}{j!}$ for integers $r\geq 0, n\geq 1$. For different values of parameter $r$, this family provides polynomials which are of great interest. Hajir conjectured that for integers $r\geq 0$ and $n\geq 1$, $L_{n}^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$, the alternating group on $n$ symbols. Extending earlier results of Schur, Hajir, Sell, Nair and Shorey, we co…
We prove that the generalised Laguerre polynomials $L_{n}^{(α)}(x)$ with $0\le \al\le 50$ are irreducible except for finitely many pairs $(n, \al)$ and that these exceptions are necessary. In fact it follows from a more general statement.
Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{ρ_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2iπ/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimur…
A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Define ${\mathcal F}(x):=x^m+Ax^{m-1}+B$. In this article, we determine sets of conditions on $m$, $A$, and $B$, such that the power-compositional trinomial ${\mathcal F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$. Furthermore, we prove the actual existence of infinite families of such trinomials ${\mathcal F}(x)$.
In the first article of this series we have presented the history of auxiliary primes from Legendre's proof of the quadratic reciprocity law up to Artin's reciprocity law. We have also seen that the proof of Artin's reciprocity law consists of several steps, the first of which is the verification of the reciprocity law for cyclotomic extensions. In this article we will show that this step can be identified with one of Dedekind's proofs of the irreducibility of the cyclotomic polynomial.
It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In this paper, we give a general criterion for guaranteeing the same conclusion under much more general coefficient distributions, allowing them to be nonuniformly and dependently distributed over arbitrary sets.
We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.
Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
Institute of Science Tokyo（东京科学大学）；Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，联合研究实验室8524，保罗·潘勒瓦实验室）；TU Graz, Institute of Analysis and Number Theory（格拉茨工业大学，分析与数论研究所）
We consider the question of certifying that a polynomial in ${\mathbb Z}[x]$ or ${\mathbb Q}[x]$ is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv.~that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we…
The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$, and $u$ would mean the non-existence of perfect cuboids. This irreducibility is conjectured and then verified numerically for approximately 10000 instances of $a$, $b$, and $u$.
Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine when $K$ is a number field. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.
Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization of T^{N,chi}_{p,k}(x) mod l as k varies, and we explicitly calculate those factorizations for N=1 and small l. These factorizations are used to deduce the irreducibility of certain T^{1,1}_{q,k}(x) from the irreducibility of T^{1,1}_{2,k}(x).
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is …
Let P_nk(x) denote the sum of the lowest k+1 terms in the expansion of (1+x)^n. We investigate the irreducibility of P_nk(x) and more general univariate polynomials related to it. Polynomials P_nk(x) naturally arise in Schubert calculus, see ArXiv e-print math.RT/0409329 by I.Scherbak.
In this article we study the irreducibility of polynomials of the form $x^n+ε_1 x^m+p^kε_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the form $x^n+ε_1x^m+ε_2$, where $ε_i\in \{\, -1,+1\,\}$. This corrects few of the existing results of W. Ljuggren's on $x^n+ε_1x^m+ε_2$.
