Shanta Laishram, T. N. Shorey
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.
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A generalization of a fourth irreducibility theorem of I. Schur
本文推广 I. Schur 于 1929 年提出的第四不可约性定理，将多项式首项系数条件由 $|a_n|=1$ 放宽到更大的整数，并证明所得结论具有最佳性。
We present an elementary proof that the Schur polynomial corresponding to an increasing sequence of exponents (c_0,..., c_{n-1}) with c_0 = 0 is irreducible over every field of characteristic p whenever the numbers d_i = c_{i+1} - c_i are all greater than 1, not divisible by p, and satisfy gcd(d_i, d_{i+1}) = 1 for every i.
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。
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)…
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。
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.
On Schur's irreducibility results and generalised $ϕ$-Hermite polynomials
本文推广 Schur 关于偶次 Hermite 型多项式的不可约性定理，证明在模小素数不可约首一多项式 $ϕ$ 及容度条件下，广义 $ϕ$-Hermite 多项式在 $\mathbb{Q}$ 上不可约。
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 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…
Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values a in O such that f(a,X) is still irreducible. In this paper we study the set Red(O) of those a in O with f(a,X) reducible. We show that Red(O) is a finite set under rather weak assumptions. In particular, several results of K. Langmann, obtained by Diophantine approximation techniques, appear as special cases of some of our results. Our method is completely…
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.
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.
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…
Let p be a prime and suppose that K/F is a cyclic extension of degree p^n with group G. Let J be the F_pG-module K^*/K^{*p} of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p^i+1 for some 0<=i<n. We show that for all primes p and all 0<=i<n, there exists a field extension K/F with a summand of dimension p^i+1.
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。
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…
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。
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.
Hasse norm principle for metacyclic extensions with trivial Schur multiplier
本文证明了当 Galois 闭包的群 $G$ 为具有平凡 Schur 乘子的亚循环群时，有限可分扩张满足 Hasse 范数原理，并提供了多种相关群的新实例。
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。
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。
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 …
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