Related papers: Large Fields in Differential Galois Theory
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular…
In this preprint we present an outline of the multidimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvability by radicals, by…
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…
Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is…
For finite Galois extension fields defined by odd degree irreducible polynomials over algebraic integer ring, we observe "Reciprocity Law" through Jacobian Variety by embedding all roots of the polynomials into 2-torsion points of Jacobian…
\newcommand{\GLn}{\operatorname{GL}_n} \newcommand{\GL}{\GLn(C)} Let $F$ be a differential field with algebraically closed field of constants $C$. We prove that $F< Y_{ij}>(X_{ij})\supset F< Y_{ij}>$ is a generic Picard-Vessiot extension of…
Hilbert's irreducibility theorem plays an important role in inverse Galois theory. In this article we introduce Hilbertian fields and present a clear detailed proof of Hilbert's irreducibility theorem in the context of these fields.
We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of…
We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then…
We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and…
We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved,…
Formulations of some Grassmann-valued systems of ordinary differential equations invariant under (infinitesimal) supersymmetry transformations, including $N$-superspace extended types, are reviewed and discussed, with use of superfields.…
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 article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension…
This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral…