Irreducibility of extensions of Laguerre Polynomials

Extensions of Schur's irreducibility results

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.

On the Irreducibility of the Krawtchouck Polynomials

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…

Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{(α)} (x) $ where $α$ is a negative integer. We observe that the constant term of $L_n^{(α)} (x) $ vanishes if and only if $n \geq |α| = -α$. Therefore we assume that $α= -n-s-1$ where $s$ is a non-negative integer. Let $$ g(x) = (-1)^n L_n^{(-n-s-1)}(x) = \sum\limits_{j=0}^{n} a_j \frac{x^j}{j!} $$ and more general polynomial, let $$ G(x) = \sum\limits_{j=0}^{n} a_j b_j \frac{x^j}{j!} $$ where $b_j$ with $0 \leq j \leq n$ are integers such that $|b_0| = |b_n| = 1$. Schur was the first to prove the irreducibility of $g(x)$ for $s=0$. It has bee…

关于扩展拉盖尔多项式的不可约性

On the irreducibility of extended Laguerre Polynomials

本文研究扩展拉盖尔多项式$L_{m,\alpha}^\phi(x)$的有理不可约性，利用$\phi$-牛顿多边形与解析数论结果，证明其除有限个$m$外不可约，并给出$\alpha$取0到4时的例外情形。

Extension of Laguerre polynomials with negative arguments II

For $n \geq 3$ and $s \leq 92$, it is proved in \cite{ShSi} that, except for finitely many pairs $(n, s), G_1(x) = G_1(x, n, s) $ is either irreducible or linear factor times an irreducible polynomial. If $s \leq 30$, we determine here explicitely the set of pairs $(n, s)$ in the above assertion. This implies a new proof of the result of Nair and Shorey \cite{NaSh1} that $G_1(x)$ is irreducible for $s \leq 22$.

Schur不可约性结果的一个推广

An extension of Schur's irreducibility result

本文推广了Schur不可约性定理，证明了在特定条件下含$\phi(x)$的加权和多项式在有理数域上不可约，并推广了Schönemann判据。

A Hilbert Irreducibility Theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials

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.

I. Schur 第二不可约性定理的一个推广

An extension of a second irreducibility theorem of I. Schur

本文在关于 $n$、首一多项式 $ϕ(x)$ 及系数容度的条件下，证明了一类含 $ϕ(x)^j/(j+1)!$ 的多项式在 $\mathbb{Q}$ 上不可约，从而推广 Schur 的第二不可约性定理。

Certifying Irreducibility in Z[x]

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…

广义杜马斯不可约性准则

A generalized Dumas irreducibility criterion

将经典杜马斯不可约性结果推广到任意秩的Krull赋值域上的多项式，证明其分解定理及因子次数下界，统一并改进了多个已知不可约性结果。

多变量多项式在有限域上绝对不可约性的新判据

A new criterion for the absolute irreducibility of multivariate polynomials over finite fields

本文提出了一种新的判据，用于判断有限域上由多变量多项式定义的超曲面的绝对不可约性，无需测试基域或扩展域的不可约性，仅需多变量GCD计算和平方自由性质。

Bertini irreducibility theorems over finite fields

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.

某些有限伽罗瓦扩张上阿贝尔簇的秩增长

Rank growth of abelian varieties over certain finite Galois extensions

该研究证明了满足特定条件的有限伽罗瓦扩张可迫使阿贝尔簇实现秩增长，结合有限群表示论得到秩增长显式下界，还将结果应用于雅可比簇并探讨其与无限秩猜想的联系。

素数与整系数多项式分解

Prime numbers and factorization of polynomials

本文研究整系数多项式不可约因子数量的上界，通过分析多项式在大整数点处的素因子分解及复平面上根的位置，推导出若干经典不可约性判别准则，并扩展至任意域上的双变量多项式。

Irreducibility and r-th root finding over finite fields

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of characteristic $p$ (equivalently, constructing the bigger field $\mathbb{F}_{q^{r^e}}$). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its …