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

Abstract

‎For singularities $fin K[[x_{1},ldots,x_{n}]]$ over an algebraically closed field $K$ of arbitrary characteristic‎, ‎we introduce the finite $mathcal{S}-$determinacy under $mathcal{S}-$equivalence‎, ‎where $mathcal{S}=mathcal{R}_{mathcal{G}},~mathcal{R}_{mathcal{A}}‎, ‎~mathcal{K}_{mathcal{G}},~mathcal{K}_{mathcal{A}}$.‎ ‎It is proved that the finite $mathcal{R}_{mathcal{G}}(mathcal{K}_{mathcal{G}})-$determinacy is equivalent to the finiteness of the relative $mathcal{G}-$Milnor ($mathcal{G}-$Tjurina) number and the finite $mathcal{R}_{mathcal{A}}(mathcal{K}_{mathcal{A}})-$determinacy is equivalent to the finiteness of the relative $mathcal{A}-$Milnor ($mathcal{A}-$Tjurina) number‎. ‎Moreover‎, ‎some estimates are provided on the degree of the $mathcal{S}-$determinacy in positive characteristic‎.