Equivalent Relations For Two Variable Real Analytic Function Germs

Singularity theory is an important mathematical tool in the study of systematic sequence evolution.It's a great application for a better explanation and predictive of mutation in nature and society,in mathematics,physics,chemistry,biology,engineering,social sciences,and so on.In mathematics,the singularity theory is an important research tool.For example,S.Koike studied the relationship between theC¹ equivalence and the blow-analytic equivalence,obtained theC¹ equivalence is blow-analytic equivalence.Furthermore,the relationship between the equivalence of bi-Lipschitz and blow-analytic equivalence is studied.The bi-lipschitz equivalent is not necessarily blow-analytic equivalent by specific examples.In this paper,we use the singularity theory as a tool to further study the equivalence for two variable real analytic function germs.The first chapter introduces the basic definition of real analytic function germs equivalence.In the second chapter,we discuss the conditions of equality of polynomial coefficientsP_f,g,x.The third chapter describes bi-lipschitz equivalent invariant and blow-anal-ytic equivalent invariant,the relationship between the K-bi-Lipschitz equivalent and the C-bi-Lipschitz equivalent,and the relationship between theC^r-Aequivalent and the topological A-equivalent.The fourth chapter discusses theC^kclassification and bi–Lipschi-tz classification of the weighted homogeneous function.