Hurwitz's theorem

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

Contents 1 Hurwitz's theorem in complex analysis 1.1 References 2 Hurwitz's theorem in algebraic geometry 2.1 References 3 Hurwitz's theorem for composition algebras 3.1 References 4 Hurwitz's theorem on Riemann surfaces 4.1 References 5 Hurwitz's theorem in number theory 5.1 References

[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 (f_n) 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 f_n 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

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 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 (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.

[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 be a finite morphism of curves over an algebraically closed field, and suppose that f is tamely ramified.

Let R be the ramification divisor

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

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 are , , and , 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 then the group Aut(M) of conformal automorphisms of M satisfies

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

Here the constant is the best possible; if we replace by any number A > 5^1/2 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