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The article investigates the behaviour of the characteristic zero resolutioninvariant when transcribed suitably to the case of surfaces in positivecharacteristic. By Moh's jumping phenomenon -- or the occurrence of kangaroosingularities -- one knows that the invariant may increase, thus destroying anyinduction.  We describe in the paper how one can modify in the purely inseparable surfacecase the invariant by adding a subtle correction term so as to prohibit itsoccasional increases and to establish the induction argument again.

Alternative Invariants for the Embedded Resolution of Purely Inseparable Surface Singularities