- Hurwitz's theorem
In
mathematics , Hurwitz's theorem is any of at least five different results named afterAdolf Hurwitz .Hurwitz's theorem in complex analysis
In
complex analysis , Hurwitz's theorem roughly states that, under certain conditions, if asequence ofholomorphic function s converges uniformly to a holomorphic function oncompact set s, then after a while those functions and the limit function have the same number of zeros in anyopen disk .More precisely, let G be an
open set in thecomplex 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 anatural 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 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.
* E. C. Titchmarsh, "The Theory of Functions", second edition (Oxford University Press, 1939; reprinted 1985), p. 119.
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 curve s, 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 algebra s over Bbb{R} are Bbb{R} , mathbb{C}, mathbb H and mathbb{O}, that is thereal number s, thecomplex number s, thequaternion s and theoctonion s.References
* John H. Conway, Derek A. Smith "On Quaternions and Octonions". A.K. Peters, 2003.
* John Baez, " [http://math.ucr.edu/home/baez/octonions/oct.pdf The Octonions] ", AMS 2001.Hurwitz's theorem on Riemann surfaces
If M is a compact
Riemann surface ofgenus g ge 2, then the group Aut(M) of conformal automorphisms of M satisfiesAut(M)| le84(g-1).Note: A conformal
automorphism of M is anyhomeomorphism of M to itself that preservesorientation , andangle s 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 everyirrational number xi there are infinitely many rationals "m"/"n" such that:left |xi-frac{m}{n} ight |
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 xi such that there exist only "finitely" many rational numbers m/n such that the formula above holds.
References
*
G. H. Hardy , E. M. Wright "An introduction to the Theory of Numbers", fifth edition, Oxford science publications, 2003.
* | year=1956
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