By Aldo Andreotti, Wilhelm Stoll

ISBN-10: 354005670X

ISBN-13: 9783540056706

ISBN-10: 3540369953

ISBN-13: 9783540369950

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**Extra resources for Analytic and Algebraic Dependence of Meromorphic Functions**

**Example text**

If U ~ ¢ is open in X, let ~(U) be the ring of holomorphic functions on U. %(U) is an integral domain. , the set of elements f ¢ ~(U) fiB ~ 0 for each branch B of U. ring of ~(U). ~(U) Let O~(U) be the total quotient If U is irreducible, then V ( U ) = ~(U) - {0} and is a field called the quotient field of ~(U). and if V and U are open, the restriction map r~: maps ~(U) into Then on X. %= such that If ~ + V ~ U ~(U) ~ ~(V) ~(V) and induces a restriction map r~: ~ ( U ) - ~ V ) . { %(U),r~} and ~ = {~(U),r~} are presheaves of rings The associated sheaf ~ to ~ is the sheaf ~ = ~ X of ~erms of holomorphlc functions on X, also called the structure sheaf of X.

6. Y be complex (0penmapplng theorem of Remmert). spaces of pure dimension m and n respectively. pose that Y is locally irreducible. Then the following three statements are equivalent. The map ~ is q-flberlng with m - n = q. b) The map ~ has pure rank n. c) The map ~ is open. 13, a) and b) are equivalent. 22, b) implies c). By Theorem Now, it shall be shown that c) implies a). If n = 0, the statement if true. ,n - i. Sup- Let ~: X ~ Y be holomorphic. a) Proof. Let X and Suppose that it is true for Then it shall be proved for n.

U of a in B exists E U. A point x ¢ U ^ exists such that ~(x) ra nk x ~0~ = rank branch of X. a simple rank is a simple (X ) ~ =< n. By Lemma point Then ~ B exists rankv(x) ~ = n. Therefore of X. r a n k a @ O V IB <= n. 16 a point x of X w i t h (x)@ = n. 16 @or has (x e B I r a n k x @ O V I B strict rankx@OVIB for each B ~ . ^ Lemma = Now ~ (B ) is a As b e f o r e = n rankx@OVIB is = By A rank n and D is thin in X. < n} is t h i n and a n a l y t i c in B. Also DB = Because B is ^ open in X, r a n k x @ O ~ I B = B/~D almost able, and D = ~ D = rankx~OV B.

### Analytic and Algebraic Dependence of Meromorphic Functions by Aldo Andreotti, Wilhelm Stoll

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