Related papers: Hrushovski's Algorithm for Computing the Galois Gr…

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an…

We consider invariants of a finite group related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We…

Let us consider a linear differential equation over a differential field K. For a differential field extension L/K generated by a fundamental system of the equation, we show that Galois group according to the general Galois theory of…

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems…

Born from years of teaching undergraduate and graduate algebra courses at Chongqing University, this text is designed to introduce Galois theory while minimizing prerequisites. It seeks to reconnect the abstract machinery of modern algeba:…

We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.

For $X$ a smooth projective variety over a field $k$, we consider the problem of Galois descent for higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Th\'el\`ene and Skorobogatov to higher Brauer groups.

The Galois group of an infinite Galois extension has a natural topology, called the Krull topology, which has the important property of being profinite. It is impossible to talk about Galois representations, and hence the Langlands Program,…

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups…

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular,…

Motivated by applications of algebraic geometry, we introduce the Galois width, a quantity characterizing the complexity of solving algebraic equations in a restricted model of computation allowing only field arithmetic and adjoining…

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. The differential representation…

Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A point $P$ in projective plane is said to be Galois if the function field extension induced by the projection $\pi_P: C \dashrightarrow \mathbb P^1$ from $P$ is…

We give a complete answer to the analogue of Grothendieck conjecture on p-curvatures for q-difference equations defined over K(x), where K is any finitely generated extension of Q and q\in K can be either a transcendental or an algebraic…

We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work by CHatzidakis, Hardouin and Singer, that addresses the problem in the case without parameters. Our…

Let G=Aut_K (K(x)) be the Galois group of the transcendental degree one pure field extension K(x)/K. In this paper we describe polynomial time algorithms for computing the field Fix(H) fixed by a subgroup H < G and for computing the fixing…

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical…

Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating…

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over…

We provide more technical details about the HLIBCov package, which is using parallel hierarchical ($\H$-) matrices to identify unknown parameters of the covariance function (variance, smoothness, and covariance length). These parameters are…