Related papers: Probabilistic Galois Theory in Function Fields

For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\ldots+a_1(t)x+a_0(t)\in\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials…

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 goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove…

Let $A = a_0T^m + \sum_{j=1}^{m-1} a_j (T^{m-j}+T^{m+j}) + T^{2m}+1 \in \mathbf{Z}[T]$ be a monic reciprocal polynomial of degree $2m$ sampled randomly by selecting its coefficients $a_0,a_1,\dots,a_{m-1}$ independently according to a given…

We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We…

Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability…

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random…

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to…

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…

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more…

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…

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for…

Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those…

An irreducible polynomial $f\in\Bbb F_q[X]$ of degree $n$ is {\em normal} over $\Bbb F_q$ if and only if its roots $r, r^q,\dots,r^{q^{n-1}}$ satisfy the condition $\Delta_n(r, r^q,\dots,r^{q^{n-1}})\ne 0$, where…

Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and…

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…

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group…