Related papers: Galois groups and prime divisors in random quadrat…

For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…

In a previous article, the second author proved that the image of the Galois representation mod $\lambda$ attached to a Hilbert modular newform is large or all but finitely many primes $\lambda$, if the form is not a theta series. In this…

Let $\F_q$ be a finite field of characteristic $p>0$. We prove that, given $F(t,x)\in \F_q[t][x]$ an irreducible separable monic polynomial in the variable $x$ and a generic monic polynomial $\phi(t)$ in the variable $t$, the polynomial…

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in…

Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group…

We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because…

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…

The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $\pi$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ \rho_{\pi,\ell} \}_\ell$…

We give a classification theorem for certain four-dimensional families of geometric $\lambda$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective…

We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided…

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $\alpha \in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre…

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…

For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the…

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois…

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline…

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 provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is…