Hilbert's twenty-second problem

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Hilbert’s twenty-second problem is one of the famous 23 problems Hilbert gave in 1900. It asks how to “uniformize” analytic relations using special functions called automorphic functions.

In simple terms, the problem asks: for any algebraic relation between two complex variables, can we always find two single‑valued functions of one complex variable so that, when we substitute these two functions, the relation becomes an identity? Poincaré showed this is possible for algebraic relations using automorphic functions.

Poincaré also studied more general analytic (non‑algebraic) relations, but it isn’t clear whether the same idea can always meet extra requirements. Specifically, can the two functions be chosen so that, as the new variable runs over its natural domain, every regular point of the given analytic relation is represented? It seems there may be many exceptional points that can only be reached by the functions approaching certain limiting values.

The problem also asks about extending the idea to three or more complex variables. There are some partial results in this direction. Picard’s work on algebraic functions of two variables is an important piece of the puzzle.

Koebe made a strong advance by proving a general uniformization result: if a Riemann surface is essentially like an open piece of the sphere (every Jordan curve splits it), then it can be conformally mapped to an open subset of the sphere. This is a significant step, but the full problem Hilbert posed remains open. There has been progress from researchers such as Griffith and Bers.