Differential Galois theory

• differential galois theory

• Liouville 

 

 

historical origin

• integration in finite terms
• quadrature of second order differential equation (Fuchsian differential equation)

 

 

•  

 

 

solvable by quadratures

• basic functions : basic elementary functions
• allowed operatrions : compositions, arithmetic operations, differentiation, integration
• examples
  
  • an elliptic integral is representable by quadrature

 

 

elementary extension

• using exponential and logarithm
• elementary element

 

 

Liouville extension

• we can adjoin
  
  • integrals
  • exponentials of integrals
  • algebraic extension (generalized Liouville extension)
• an element is said to be representable by a generalized quadrature
• For\(K_{i}=K_{i-1}(e_i)\) , one of the following condition holds
  
  • \(e_i'\in K_{i-1}\), i.e.
    
  • \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
    
  • \(e_{i}\) is algebraic over \(K_{i-1}\)
    
• K/F is a Liouville extension iff the differential Galois group K over F is solvable.
  
• K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity
  

 

 

Picard-Vessiot extension

• framework for linear differential equation
• made by including solutions of DE to the base field (e.g. rational function field)
• this corresponds to the concept of the splitting fields(or Galois extensions)
• we can define a Galois group for a linear differential equation.
• examples
  
  • algebraic extension
  • adjoining an integral
  • adjoining the exponential of an integral

 

theorem

If a Picard-Vessiot extension is a Liouville extension, then the Galois group of this extension is solvable.

 

 

Fuchsian differential equation

• differential equation with regular singularities
• indicial equation
  \(x(x-1)+px+q=0\)
  

theorem

A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.

 

 

 

solution by quadrature

• Differential Galois Theory and Non-Integrability of Hamiltonian Systems
• Integrability and non-integrability in Hamiltonian mechanics
• [1]http://www.caminos.upm.es/matematicas/morales%20ruiz/libroFSB.pdf
• http://andromeda.rutgers.edu/~liguo/DARTIII/Presentations/Khovanskii.pdf
• http://www.math.purdue.edu/~agabriel/topological_galois.pdf

 

 

related items

• Class Field Theory
  
• number fields and threefolds
  
• Fuchsian 미분방정식(Fuchsian differential equation)
  

 

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Differential_Galois_theory
• http://en.wikipedia.org/wiki/Homotopy_lifting_property
• http://en.wikipedia.org/wiki/covering_space
• http://en.wikipedia.org/wiki/Field_extension

 

 

articles

• Liouvillian First Integrals of Differential Equations
  
  • Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
• Elementary and Liouvillian solutions of linear differential equations
  
  • M. F. Singer and J. H. Davenport, 1985

 

 

books

• Group Theory and Differential Equations
  
  • Lawrence Markus, 1960
• An introduction to differential algebra
  
  • Irving Kaplansky
    
• algebraic theory of differential equations
  
• http://gigapedia.info/1/galois_theory
• http://gigapedia.info/1/differential+galois+theory
• http://gigapedia.info/1/Kolchin
• http://gigapedia.info/1/ritt
• http://gigapedia.info/1/Galois'+dream
• http://gigapedia.info/1/differntial+algebra