Math 2.0

Ordinary differential equations Complex domain and special equations Problem

Hilbert's twenty-first problem

Builds on MonodromyFuchsian equationRegular singular pointRiemann–Hilbert problem

Problem

Let a1,…,ama_1, \ldots, a_m be distinct points of the Riemann sphere, let UU be the sphere with these points removed, and fix z0∈Uz_0 \in U. Let ρ ⁣:π1(U,z0)→GLn(ℂ)\rho \colon \pi_1(U, z_0) \to \mathrm{GL}_n(\C) be a group homomorphism. Does there exist a linear differential equation of Fuchsian class whose singular points are a1,…,ama_1, \ldots, a_m and whose monodromy representation is ρ\rho?

Remarks

In the form used today, the question is asked for first-order systems Y′=A(z)YY' = A(z) Y of Fuchsian type, A(z)=∑iAiz−aiA(z) = \sum_{i} \frac{A_i}{z - a_i} with constant matrices AiA_i, which have simple poles only. If one only asks for a system with regular singular points, the answer is positive; this was shown by Plemelj in 1908. Plemelj's work was long taken as a positive solution for Fuchsian systems. In 1989 Bolibrukh found a counterexample: a representation that is not the monodromy of any Fuchsian system with the given singular points. The answer is positive when ρ\rho is irreducible, a result of Bolibrukh and of Kostov. For scalar equations of order nn, extra apparent singular points are needed in general. Plemelj's approach reduces the question to a matrix Riemann–Hilbert problem on a curve through the singular points.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

Show the path as a graph
Complex numbers, object, no evidence levelComplex numbersobjectElement, definition, no evidence levelElementdefinitionFuchsian equation, concept, no evidence levelFuchsian equationconceptFunction, concept, no evidence levelFunctionconceptHolomorphic function, concept, no evidence levelHolomorphic functionconceptMonodromy, concept, no evidence levelMonodromyconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectRegular singular point, concept, no evidence levelRegular singular pointconceptContour integral, definition, no evidence levelContour integraldefinitionSet, definition, no evidence levelSetdefinitionLaplace transform, definition, no evidence levelLaplace transformdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionCauchy principal value, definition, no evidence levelCauchy principal valuedefinitionWinding number, definition, no evidence levelWinding numberdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionHeaviside step function, object, no evidence levelHeaviside step functi…objectExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionDirac delta function, object, no evidence levelDirac delta functionobjectSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionHölder continuity, definition, no evidence levelHölder continuitydefinitionSokhotski–Plemelj theorem, statement, no evidence levelSokhotski–Plemelj the…statementRiemann–Hilbert problem, problem, no evidence levelRiemann–Hilbert probl…problemHilbert's twenty-first problem, problem, no evidence levelHilbert's twenty-firs…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Element definition
  3. Fuchsian equation concept
  4. Function concept
  5. Holomorphic function concept
  6. Monodromy concept
  7. Proposition concept
  8. Real numbers object
  9. Regular singular point concept
  10. Contour integral definition
  11. Set definition
  12. Laplace transform definition
  13. Implication definition
  14. Predicate definition
  15. Cauchy principal value definition
  16. Winding number definition
  17. Ordered pair definition
  18. Heaviside step function object
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Dirac delta function object
  23. Subset definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Function definition
  27. Metric space definition
  28. Lipschitz map definition
  29. Hölder continuity definition
  30. Sokhotski–Plemelj theorem statement
  31. Riemann–Hilbert problem problem
  32. Hilbert's twenty-first problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_twenty-first_problem
About this page
Tag7RHW0KA5RY
URIm2://7RHW0KA5RY
Kindproblem
WritingW0 stub
MSC34M50, 34M35
StatusThis copy of the site has no build data. Run m2 status 7RHW0KA5RY.
AboutMonodromy, Fuchsian equation
UsesMonodromy, Fuchsian equation, Regular singular point, Riemann–Hilbert problem
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.