A Measure of Integrity for Local Analytic Algebras

We will complete the study [IJ, [I2] on orders of elements of a local analytic algebra. Let (X, 0X) be a complex space and ( 0 , m ) =( ® x.& %) its local algebra at $ &X. (9 can be expressed as 0 = C [x] /I for some ideal / of the algebra C {%} of convergent power series in x= (x^ • • • , Xm). We define three kinds of orders for f^O. algebraic order: ^ (/)= ^ (/)"•= sup {/?: /^ T0.] = sup {the degree of the lowest non-zero homogeneous term of /: / is a representative of f in C(x}} reduced order ([ReJ, cf. [S]): y(/):=lim v(/*)/A J-»oo