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We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{X}$ an irreducible affine algebraic variety…

Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following:…

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois…

We present an algorithm that determines the Galois group of linear difference equations with rational function coefficients.

We obtain a criterion for when the specialization of the iterated Galois group for a post-critically finite (PCF) rational map is as large as possible, i.e., it equals the generic iterated Galois group for the given map.

Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be…

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…

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this…

Galois categories can be viewed as the combinatorial analog of Tannakian categories. We introduce the notion of pre-Galois category, which can be viewed as the combinatorial analog of pre-Tannakian categories. Given an oligomorphic group…

We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.

We introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our…

An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…

The extended L\"uroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In…

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper we give sufficient conditions that a given finite group $G$ occurs infinitely often as a Galois group over the rationals…

In this note, we give a practical solution to the problem of determining the maximal period of matrix generators of pseudo-random numbers which are based on an integer-valued unimodular matrix of size NxN known as MIXMAX and arithmetic…

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups…

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$,…

This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $\l$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the…

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra…