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HURWITZ'S THEOREM IN COMPLEX ANALYSIS


In Complex Analysis , Hurwitz's theorem roughly states that, under certain conditions, if a Sequence of Holomorphic Function s Converges Uniformly to a holomorphic function on Compact Set s, then after a while those functions and the limit function have the same number of Zeros in any Open Disk .

More precisely, let G be an Open Set in the Complex Plane , and consider a sequence of holomorphic functions (f_n) which converges uniformly on compact subsets of G to a holomorphic function f. Let D(z_0,r) be an open disk of center z_0 and radius r which is contained in G together with its Boundary . Assume that f(z) is non-zero on the disk boundary. Then, there exists a Natural Number N such that for all n greater than N the functions f_n and f have the same number of zeros in D(z_0,r).

The requirement that f be nonzero on the disk boundary is necessary. For example, consider the disk of center zero and radius 1, and the sequence

:f_n(z) = z-1+ rac{1}{n}

for all z. It converges uniformly to f(z)=z-1 which has no zeros inside of this disk, but each f_n(z) has exactly one zero in the disk, which is 1-1/n.

This result holds more generally for any Bounded Convex Set s but it is most useful to state for disks.

An immediate consequence of this theorem is the following Corollary . If G is an open set and a sequence of holomorphic functions (f_n) converges uniformly on compact subsets of G to a holomorphic function f, and furthermore if f_n is not zero at any point in G, then f is either identically zero or also is never zero.


References


  • John B. Conway. ''Functions of One Complex Variable I''. Springer-Verlag, New York, New York, 1978.






HURWITZ'S THEOREM IN ALGEBRAIC GEOMETRY


In algebraic geometry, the result referred to as Hurwitz's theorem is an index theorem which relates the Degree of a Branched Cover of Algebraic Curves , the genera of these curves and the behaviour of f at the branch points.

More explicitly, let f: X ightarrow Y be a finite morphism of curves over an algebraically closed field, and suppose that f is tamely ramified.

Let R be the ramification divisor R= \sum_{P \in X} (e_{P}-1) P, where e_{P} denotes the ramification index of f at P. Let n:= deg f, and let g(X), g(Y) denote the genus of X, Y respectively.

Then Hurwitz's theorem states that

2 g(X) - 2 = n (2 g(Y) - 2) + deg R .


References

  • R. Hartshorne, Algebraic Geometry, Springer, New York 1977



HURWITZ'S THEOREM FOR COMPOSITION ALGEBRAS


In this context, Hurwitz's theorem states that the only Composition Algebras over \Bbb{R} are \Bbb{R} , \mathbb{C}, \mathbb H and \mathbb{O}, that is the Real Number s, the Complex Number s, the Quaternion s and the Octonion s.


References


  • John H. Conway, Derek A. Smith ''On Quaternions and Octonions''. A.K. Peters, 2003.



HURWITZ'S THEOREM IN NUMBER THEORY

In number theory, the Hurwitz's theorem states that for every irrational number \xi there are infinitely many rationals ''m''/''n'' such that