scholarly journals Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations

2014 ◽  
Vol 2015 (7) ◽  
pp. 1733-1793 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov ◽  
Michael F. Singer
Keyword(s):  
Differential Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

scholarly journals DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 16 (1) ◽  
pp. 59-119 ◽  
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin ◽  
Michael Wibmer
Keyword(s):  
Differential Equations ◽  
Characteristic Zero ◽  
Galois Theory ◽  
Structure Theorem ◽  
Special Functions ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

scholarly journals Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

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