Inner rates of finite morphisms

Let ( X , 0) be a complex analytic surface germ embedded in (Cⁿ,0) ( C n , 0 ) with an isolated singularity and Φ =(g,f):(X,0) (C²,0) Φ = ( g , f ) : ( X , 0 ) ⟶ ( C 2 , 0 ) be a finite morphism. We define a family of analytic invariants of the morphism Φ Φ , called inner rates of Φ Φ . By means of the inner rates we study the polar curve associated with the morphism Φ Φ when fixing the topological data of the curve (gf)⁻¹(0) ( g f ) - 1 ( 0 ) and the surface germ ( X , 0), allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function f:(X,0) (C,0) f : ( X , 0 ) ⟶ ( C , 0 ) . The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ ( X , 0) and of the curve (gf)⁻¹(0) ( g f ) - 1 ( 0 ) .