Differential Galois theory

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introduction

  • differential galois theory
  • Liouville
  • 2008년 12월 9일 MCF 'differential Galois theory'



historical origin

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



solution by quadrature

  • 일계 선형미분방정식\(\frac{dy}{dx}+a(x)y=b(x)\)\(y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C\)
  • \(y''-2xy'=0\)\(y=\int e^{x^2}\, dx\)
  • note that the integral of an exponential naturally shows up in expression solutions



differential field

  • a pair \((F,\partial)\) such that
    • \(\partial(a+b)=\partial a+\partial b\)
    • \(\partial(ab)=(\partial a)b+a(\partial b)\)
  • \(C_F=\ker \partial\)



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

  • it is allowed to take exponentials and logarithms to make a field extension
  • elementary element
  • difference between Liouville extension
    • exponential+ integral <=> differentiation + exponential of integral
    • in elementary extension, we are not allowed to get an integrated element



Liouville extension

  • an element is said to be representable by a generalized quadrature
  • we can capture these properties using the concept of Liouville extension
  • to get a Liouville extension, we can adjoin
    • integrals
    • exponentials of integrals
    • algebraic extension (generalized Liouville extension)
      • from these we can include the following operations
        • exponential
        • logarithm
  • For\(K_{i}=K_{i-1}(e_i)\) , one of the following condition holds
    • \(e_i'\in K_{i-1}\), i.e. \(e_i=\int e_i'\in K_i\)
    • \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
    • \(e_{i}\) is algebraic over \(K_{i-1}\)
  • remark on exponentiation
    • Let \(a,a'\in F\). Is \(b=e^a\in K\) where K is a Liouville extension?
    • \(b'=a' e^a=a'b\) implies \(a'=\frac{b'}{b}\in F\).
    • the exponential of the integral of a' i.e. \(e^{\int a'}=e^a+c\) must be in the Liouville extension. So \(b=e^a\in K\).
  • remark on logarithm
    • \(b=\log a\) is the integral of \(a'/a\in F\). So \(b\in K\)
  • a few result
    • 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
  • field extension is made by including solutions of DE to the base field (e.g. rational function field)
  • consider monic differential equations over a differential field F\(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
  • \((E,\partial_E)\supseteq (F,\partial_F)\) is a Picard-Vessiot extension for \(\mathcal{L}\) if
    • E/F is generated by n linear independent solution to \(\mathcal{L}\), i.e. adjoining basis of \(V=\mathcal{L}^{-1}(0)\) to F
    • \(C_E=C_F\), \(\partial_E\mid_F=\partial_F\)
  • this corresponds to the concept of the splitting fields(or Galois extensions)
  • examples
    • algebraic extension
    • adjoining an integral
    • adjoining the exponential of an integral
  • we can define a Galois group for a linear differential equation\(\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}\)
    • the action of an element of the Galois group is determined by its action on a basis of V

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



related items




encyclopedia



expositions

  • Singer, M. F., and J. H. Davenport. ‘Elementary and Liouvillian Solutions of Linear Differential Equations’. In EUROCAL ’85, edited by Bob F. Caviness, 595–96. Lecture Notes in Computer Science 204. Springer Berlin Heidelberg, 1985. http://link.springer.com/chapter/10.1007/3-540-15984-3_335.


articles

  • Sanchez, Omar Leon, and Joel Nagloo. “On Parameterized Differential Galois Extensions.” arXiv:1507.06338 [math], July 22, 2015. http://arxiv.org/abs/1507.06338.
  • Maurischat, Andreas. “A Categorical Approach to Picard-Vessiot Theory.” arXiv:1507.04166 [math], July 15, 2015. http://arxiv.org/abs/1507.04166.
  • Hardouin, Charlotte, and Alexey Ovchinnikov. ‘Calculating Galois Groups of Differential Equations with Parameters’. arXiv:1505.07068 [math], 26 May 2015. http://arxiv.org/abs/1505.07068.
  • Blázquez-Sanz, David, Juan José Morales-Ruiz, and Jacques-Arthur Weil. ‘Differential Galois Theory and Lie Symmetries’. arXiv:1503.09023 [math], 31 March 2015. http://arxiv.org/abs/1503.09023.
  • Mitschi, Claude. “Some Applications of the Parameterized Picard-Vessiot Theory.” arXiv:1503.01361 [math], March 4, 2015. http://arxiv.org/abs/1503.01361.
  • Crespo, Teresa, Zbigniew Hajto, and Elzbieta Sowa-Adamus. ‘Galois Correspondence Theorem for Picard-Vessiot Extensions’. arXiv:1502.08026 [math], 27 February 2015. http://arxiv.org/abs/1502.08026.
  • Singer, Michael F. ‘Liouvillian First Integrals of Differential Equations’. Transactions of the American Mathematical Society 333, no. 2 (1 October 1992): 673–88. doi:10.2307/2154053.

books

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