Related papers: Generalized Artin-Schreier polynomials
Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $f(x)$. We investigate the arithmetic…
The theorem proved in this note, although elementary, is related to a certain misconception. If $K$ is a field, $f\in K[X]$ is separable and irreducible over $K$, and $g$ is a polynomial dividing $f$, whose coefficients lie in some finite…
Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length…
In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We…
Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann…
This is an updated version of ANT-0253. Let F be a number field with absolute Galois group G. We associate, to each continuous, solvable C-representation of G of GO(4)-type, an automorphic form P of GL(4)/F with the same L-function. As a…
The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all…
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…
Let $\tilde{f}(X)\in\mathbb{Z}[X]$ be a degree-$n$ polynomial such that $f(X):=\tilde{f}(X)\bmod p$ factorizes into $n$ distinct linear factors over $\mathbb{F}_p$. We study the problem of deterministically factoring $f(X)$ over…
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first…
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first…
We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental…
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. To any irreducible smooth…
For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We deduce a necessary and sufficient…
We describe the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler…
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…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer…
We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly…
The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime…