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

Hengxing, L.; Jingwen, L‎. ‎

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

Document Type : Research Paper

Authors

L. Hengxing ¹
L‎. ‎ Jingwen ²

¹ School of Mathematics and Statistics,Wuhan University, Wuhan, assistance professor

² School of Mathematics and Statistics‎, ‎Wuhan University‎, ‎P.O‎. ‎Box 430072‎, ‎Wuhan‎, ‎People's Republic of China

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.

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