Hurwitz's theorem

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

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 $D(z_{0},r)$ be an open disk of center $z_{0}$ and radius $r$ which is contained in $G$ together with its boundary. Assume that $f(z)$ 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 $D(z_{0},r).$

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 $f_{n}(z)$ 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.

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.
• Solomentsev, E.D. (2001) [1994], "Hurwitz theorem", Encyclopedia of Mathematics, EMS Press

Hurwitz's theorem at PlanetMath.

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 $e_{P}$ 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.

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 ${\mathbb {R}}$ are ${\mathbb {R}}$, $\mathbb {C}$, $\mathbb {H}$ and $\mathbb {O}$, that is the real numbers, the complex numbers, the quaternions and the octonions.

References

• Hurwitz, A. (1898). "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln (On the composition of quadratic forms of arbitrary many variables)". Nachr. Ges. Wiss. Göttingen (in German): 309–316. JFM 29.0177.01.
• John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.
• John Baez, The Octonions, AMS 2001.

Hurwitz's theorem on Riemann surfaces

Main article: Hurwitz's automorphisms theorem

If $M$ is a compact Riemann surface of genus $g\geq 2,$ then the group Aut(M) of conformal automorphisms of M satisfies $|\operatorname {Aut} (M)|\leq 84(g-1).$

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

References

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

Hurwitz's theorem in number theory

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

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

The hypothesis that ξ is irrational cannot be omitted. Moreover the constant $\scriptstyle {\sqrt {5}}$ is the best possible; if we replace $\scriptstyle {\sqrt {5}}$ by any number $\scriptstyle A>{\sqrt {5}}$ and we let $\scriptstyle \xi =(1+{\sqrt {5}})/2$ (the golden ratio) then there exist only finitely many rational numbers m/n such that the formula above holds.

References

• Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximation of irrational numbers by rational numbers)". Mathematische Annalen (in German). 39 (2): 279–284. doi:10.1007/BF01206656. JFM 23.0222.02.
• G. H. Hardy, Edward M. Wright, Roger Heath-Brown, Joseph Silverman, Andrew Wiles (2008). "Theorem 193". An introduction to the Theory of Numbers (6th ed.). Oxford science publications. p. 209. ISBN 0199219869.{{cite book}}: CS1 maint: multiple names: authors list (link)
• LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR0080682