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2. Complexity


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      Problem 2.1.

      [Julia Knight] Let K be an algebraically closed field of characteristic 0, and let G be an ordered divisible group. Consider the Hahn field, defined as follows: K((G)) = \left\{\sum_{g\in S}a_g t^g \mid S\subseteq G \text{ well-ordered}, a_g\in K\right\}


      If K and G are computable and f\in K((G))[X], how complex is it to determine a root of f? At each step, the procedure should produce the next term in a root of f, based on the a_g in the input coefficients of f. For each step \alpha, how many jumps over these coefficients are needed to do step \alpha?

          Cite this as: AimPL: Definability and decidability problems in number theory, available at http://aimpl.org/definedecide.