October 30, 2013
In this article, we consider the inverse Galois problem for parameterized differential equations over k((t))(x) with k any field of characteristic zero and use the method of patching over fields due to Harbater and Hartmann. As an application, we prove that every connected semisimple k((t))-split linear algebraic group is a parameterized Galois group over k((t))(x).
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March 22, 2016
This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated...
January 27, 2015
In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables).
February 18, 2005
We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic...
January 20, 2014
We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form \[ \tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0, \] where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under ...
December 3, 2019
We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.
July 24, 2007
A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field extension.
May 5, 2014
Let $C \langle \boldsymbol{t} \rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots, t_l)$ over an algebraically closed field $C$ of characteristic zero. In this article we present an explicit linear parameter differential equation over $C \langle \boldsymbol{t} \rangle$ with differential Galois group $\mathrm{SL}_{l+1}(C)$ and show that it is a generic equation in the following sense: If $F$ is an algebraically closed dif...
September 18, 2016
Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound criterion for the differential Galois group $G(C)$ of a matrix parameter differential equation $\partial(\boldsymbol{y})=A(\boldsymbol{t})\boldsymbol{y}$ over $C \langle t_1, \dots t_l\rangle$ and we prove that every connected linear algebraic grou...
March 22, 2024
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 pullback is itself internal to the constants. To do so, we develop, using model-theoretic Galois theory and differential algebra, a connection between internality of the pullback and the splitting of a short exact sequence of algebraic Gal...
May 17, 2019
This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over e...