Complex Analysis and Algebraic Geometry by Grauert G. (ed.)

By Grauert G. (ed.)

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If f and g are functions defined in neighborhoods U and V of x and if f (y) = g(y) for all y in some third neighborhood W of x with W C U n V, then we say that f and g are equivalent at x. The equivalence class consisting of all functions equivalent to f at x is called the germ of f at x. The set of germs of complex valued functions at x is clearly an algebra over the complex field, with the algebra operations defined in the obvious way. In fact, this algebra can be described as the inductive limit lim F(U), where F(U) is the algebra of complex valued functions on U, and the limit is taken over the directed set consisting of neighborhoods of x.

G1, ... , gm. Therefore, f belongs to the ideal generated by g1, ... , gm, and we conclude that this set of elements generates Z.

0, zn) has vanishing order k for some k, that is, if it does not vanish identically in any neighborhood of zn = 0. Thus, if f is not identically 0 in a neighborhood of z = 0, then coordinates can be chosen for CCn so that f has finite vanishing order in zn at 0. A germ f E Ho is said to have vanishing order k in zn if it has a representative in some neighborhood of 0 which has vanishing order k in zn at 0. 1 Proposition. If f is holomorphic in a neighborhood U of 0 in CCn and has vanishing order k in zn at 0, then there is a polydisc 0 (0, r') x 0 (0, rn) such that, for each z' E 0(0, r'), as a function of zn, f (z', zn) has exactly k zeroes in 0(0, rn), counting multiplicity, and no zeroes on the boundary of 0(0,rn).

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