The finite $S$-􀀀determinacy of singularities in positive characteristic $S=R_G,R_A‎, ‎K_G,K_A$

For singularities f 2 K[[x1,..., xn]] over an algebraically closed eld K of arbitrary characteristic, we introduce the nite S??determinacy under S??equivalence, where S = RG; RA; KG; KA. It is proved that the nite RG(KG)??determinacy is equivalent to the niteness of the relative G??Milnor (G??Tjurina) number and the nite RA(KA)??determinacy is equivalent to the niteness of the relative A??Milnor (A??Tjurina) number. Moreover, some estimates are provided on the degree of the S??determinacy in positive characteristic.