Related papers: Towards van der Waerden's conjecture

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such…

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2…

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such…

In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular,…

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include…

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field…

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…

A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree…

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…

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…

We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational…

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r…

In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is…

Let $n\ge 3$ be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree $n$ with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated…

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…

We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group,…

We show that there are at most $O_{n,\epsilon}(H^{n-2+\sqrt{2}+\epsilon})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973…

In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is…

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 $\mu$ 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…