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We establish an algorithm for resolution of singularities of an idealisticfiltration in dimension 3 (at the local level) in positive characteristic,incorporating the method recently developed by Benito-Villamayor into ourframework. Although (a global version of) our algorithm only implies embeddedresolution of surfaces in the smooth ambient space of dimension 3, a classicalresult known before, we introduce some new invariant which effectively measureshow much singularities are improved in the process of our algorithm and whichstrictly drops after each blow up. This is in contrast to the well-knownAbhyankar-Moh pathology of the increase of the residual order under blow up andthe phenomenon of the "Kangaroo" points observed by Hauser.

Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor