Hurwitz's theorem

From Wikipedia, the free encyclopedia

In mathematics, Hurwitz's theorem is any of at least five different results named after Adolf Hurwitz.

Contents

[edit] Hurwitz's theorem in complex analysis

In complex analysis, Hurwitz's theorem roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, 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 (fn) which converges uniformly on compact subsets of G to a holomorphic function f. Let D(z0,r) be an open disk of center z0 and radius r which is contained in G together with its boundary. Assume that f(z) has no zeros on the disk boundary. Then, there exists a natural number N such that for all n greater than N the functions fn and f have the same number of zeros in D(z0,r).

The requirement that f have no zeros 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+\frac{1}{n}

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

This result holds more generally for any bounded convex sets 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 (fn) converges uniformly on compact subsets of G to a holomorphic function f, and furthermore if fn is not zero at any point in G, then f is either identically zero or also is never zero.

[edit] References

  • John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
  • E. C. Titchmarsh, The Theory of Functions, second edition (Oxford University Press, 1939; reprinted 1985), p. 119.

This article incorporates material from Hurwitz's theorem on PlanetMath, which is licensed under the GFDL.

[edit] Hurwitz's theorem in algebraic geometry

Main article: Riemann-Hurwitz formula

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 \rightarrow 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 eP 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

2g(X) − 2 = n(2g(Y) − 2) + deg R.

[edit] References

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

[edit] 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 numbers, the complex numbers, the quaternions and the octonions.

[edit] References

  • John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.
  • John Baez, The Octonions, AMS 2001.

[edit] Hurwitz's theorem on Riemann surfaces

Main article: Hurwitz's automorphisms theorem

If M is a compact Riemann surface of genus g \ge 2, then the group Aut(M) of conformal automorphisms of M satisfies |Aut(M)| \le84(g-1).

Note: A conformal automorphism of M is any homeomorphism of M to itself that preserves orientation, and angles along with their senses (clockwise/counterclockwise.)

[edit] References

  • H. Farkas and I. Kra, "Riemann Surfaces", 2nd ed., Springer, 2004, § V.1, p. 257ff.

[edit] Hurwitz's theorem in number theory

In the field of Diophantine approximation, Hurwitz's theorem states that for every irrational number ξ there are infinitely many rationals m/n such that

\left |\xi-\frac{m}{n}\right |<\frac{1}{\sqrt{5}\, n^2}.

Here the constant \sqrt{5} is the best possible; if we replace \sqrt{5} by any number A > 51/2 then there exists at least one irrational ξ such that there exist only finitely many rational numbers m/n such that the formula above holds.

[edit] References

  • G. H. Hardy, E. M. Wright An introduction to the Theory of Numbers, fifth edition, Oxford science publications, 2003.
  • LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR0080682 
Languages