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  "body": "\n## Problem\n\nLet $\\Gamma \\subset \\C$ be a smooth simple closed curve, oriented counterclockwise, with interior domain $D^+$ and exterior domain $D^-$.\nLet $n \\ge 1$ and let $G \\colon \\Gamma \\to \\mathrm{GL}_n(\\C)$ be a [[Hölder continuous|012PBA0FMD]] map into the [[invertible|0EB9N0S33C]] complex $n \\times n$ matrices.\nLet $g \\colon \\Gamma \\to \\C^n$ be Hölder continuous.\nFind a function $\\Phi$ with values in $\\C^n$ that is [[holomorphic|0NAHXR1FRH]] on $D^+$ and on $D^-$.\nIt may have at most a pole of prescribed order at $\\infty$.\nIt must have continuous boundary values $\\Phi^+$ from $D^+$ and $\\Phi^-$ from $D^-$.\nThese must satisfy\n$$\\Phi^+(t) = G(t)\\, \\Phi^-(t) + g(t), \\quad t \\in \\Gamma.$$\nThe *homogeneous* problem is the case $g = 0$.\n\n## Remarks\n\nThis is the vector form of the [[Riemann–Hilbert problem|RY8YQ9JZF9]]; for $n = 1$ it is the scalar problem, which is solved in closed form with logarithms and Cauchy integrals.\nFor $n \\ge 2$ the scalar method fails, because the values $G(t)$ need not commute.\n\nThe general theory rests on a factorization $G = G_+\\, \\Lambda\\, G_-$ on $\\Gamma$, where $G_+$ extends holomorphically and invertibly to $D^+$, $G_-$ does the same on $D^-$ including $\\infty$, and $\\Lambda(t) = \\operatorname{diag}(t^{\\kappa_1}, \\ldots, t^{\\kappa_n})$ when $0 \\in D^+$.\nSuch a factorization exists for every Hölder continuous invertible $G$, and the integers $\\kappa_1 \\ge \\cdots \\ge \\kappa_n$, the partial indices, are unique.\nTheir sum is the [[winding number|0X7SBS8E0F]] of $\\det G$ about $0$.\nThe number of solutions of the homogeneous problem and the solvability conditions of the inhomogeneous problem are expressed by the partial indices, not by their sum alone.\nThe same structure appears for matrix symbols on the real line in the [[matrix Wiener–Hopf factorization|2M672PP7QR]].\nNo general method computes the partial indices or the factors of a given $G$ in closed form.\n\nExample: if $G$ is diagonal, the problem splits into $n$ scalar problems, and the partial indices are the winding numbers of the diagonal entries.\nIf $G$ is triangular, the problem is solved by a chain of scalar problems.\n\nThe problem is linked to Hilbert's twenty-first problem, which asks for a linear system of differential equations with regular singular points and prescribed monodromy.\nPlemelj gave a positive answer for regular singular systems in 1908 through this problem.\nBolibruch showed in 1989 that the version asking for Fuchsian systems has a negative answer in general.\nMatrix Riemann–Hilbert problems on contours with prescribed jumps are also the basis of the nonlinear steepest descent method for integrable systems and random matrices.\n\n## Explanations\n\n### L1 human\n\nOne looks for a vector of functions that is holomorphic on both sides of a closed curve.\nAcross the curve the vector jumps: the values from inside equal a given matrix times the values from outside.\nFor a single function this is solved exactly, but with matrices the order of multiplication matters, and the problem becomes much harder.\nIts answer is controlled by a list of integers, the partial indices, which are hidden inside the jump matrix.\n",
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      "first_stated": "1900",
      "notes": "Hilbert listed it as the 21st of his problems in 1900. Riemann had studied the question in 1857.",
      "posed_by": "David Hilbert"
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