Related papers: From Polynomials to Databases: Arithmetic Structur…

In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to…

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…

I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational…

Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois group of $g_4(x)$ and $g_6(x)$, respectively. Building upon known characterizations of…

We give a complete description of the normal subgroups of arboreal Galois groups of Belyi maps. The normal groups form a unique chief series. We also carefully compute the discriminate of the iterate of a polynomial minus an algebraic…

This article is interested in pullbacks under the logarithmic derivative of algebraic ordinary differential equations. In particular, assuming the solution set of an equation is internal to the constants, we would like to determine when its…

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $f\in H^1(K,T)$ with discriminant-like invariant ${\rm inv}(f)\le X$ as $X\to\infty$. Specifically, Poisson summation produces a canonical…

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We then apply this to constructively test if solutions of linear q-difference…

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…

We compute all signatures of $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ which classify all orientation preserving actions of the groups $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ on compact, connected, orientable surfaces…

We study the determinant of certain etale sheaves constructed via middle convolution in order to realize special linear groups regularly as Galois groups over the rationals.

In this work, we present a mathematical formulation for machine learning of (1) functions on symmetric matrices that are invariant with respect to the action of permutations by conjugation, and (2) functions on point clouds that are…

Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…

We prove that for a system of indeterminates (X_a) indiced by the P^2(2), the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (X_a) is one of the orbits of it. We…

Galois/monodromy groups attached to parametric systems of polynomial equations provide a method for detecting the existence of symmetries in solution sets. Beyond the question of existence, one would like to compute formulas for these…

We address two interrelated problems concerning the permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $\varphi(x)\in\mathbb{C}[y_1,\cdots,y_k][x]$ over…

We present a simple method for the construction of polynomials with cyclic Galois groups, hoping to encourage a reader with some background in algebra to make computations of his/her own.

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the…

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ for $Q$ some integral quadratic form of…