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ABSTRACT: In the model theory of valued fields, one of main themes is the Ax-Kochen-Ershov principle. Roughly speaking, it says that the theory of a henselian valued field of characteristic 0 is determined by the theories of its value group and residue field. For the unramified case, it was proved by Ax-Kochen and independently by Ershov. However, for the finitely ramified case, the residue field is not enough to determine the theory of a given valued field. For finitely ramified case, it is necessary to consider theories of residue rings of all finite orders, and furthermore, it is enough to consider a residue ring of a single order, proved by Basarab, Lee -L., and Anscombe-Dittmann-Jahnke. In this talk, we introduce our recent result to improve an upper bound of the order of residue ring, which guarantee the Ax-Kochen-Ershov principle for the finitely ramified case. Especially, our improved bound is linear in the tame ramification index (and the previous known bounds are all quadratic in the tame ramification index). This is a joint work with Wan Lee.