Possible Values for Preimage Count in Meromorphic Functions on Riemann Sphere?

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In summary, the conversation involved discussing basic questions about the Riemann Sphere, particularly involving meromorphic functions. The first question was to find all meromorphic functions that satisfy f(f)=f, leading to the possibility of only the identity map or a constant map. However, the existence of f^-1 rules out the latter. The second question involved determining the possible values of n for a meromorphic function with a preimage of c containing n elements. It was suggested to use Laurent expansions to prove that f(f)=f only when degf=1 or 0, leading to the conclusion that f must be either the identity map or a constant map.
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Ant farm
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Hi there,
working on some basic questions involving the Riemann Sphere(sigma): C union infinity

firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...
was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??
stuck here, any hints would be great!
thank you.
 
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  • #2
Ant farm said:
Hi there,
working on some basic questions involving the Riemann Sphere(sigma): C union infinity

firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...
was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...


f^2=f does not imply f=f^-1. Firstly, f^-1 need not exist, indeed cannot exist, unless f=Id. There are also more maps than just Id that satisfy f=f^-1 (or f^2=Id).



secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??
stuck here, any hints would be great!
thank you.


My first thoughts are that meromorphic functions have Laurent expansions.
 
  • #3
it seems you have proved that f(f) = f implies degf = 1 or 0.that does sound as if f is id or constant, can you prove that?
 

FAQ: Possible Values for Preimage Count in Meromorphic Functions on Riemann Sphere?

What is a meromorphic function?

A meromorphic function is a complex-valued function that is defined and holomorphic (analytic) on the entire complex plane except for a set of isolated points, where it may have poles. In other words, it is a function that is a ratio of two analytic functions and is defined everywhere except for a finite number of points.

What is the difference between a meromorphic function and a holomorphic function?

A meromorphic function differs from a holomorphic function in that it may have poles, whereas a holomorphic function is defined and analytic on its entire domain. Additionally, a holomorphic function must also satisfy the Cauchy-Riemann equations, while a meromorphic function does not necessarily have to.

How are poles and zeros related in a meromorphic function?

In a meromorphic function, poles and zeros are related by the principle of the argument principle. This states that the number of zeros of a meromorphic function inside a closed curve is equal to the number of poles inside the same curve, counted with multiplicities.

What is the importance of meromorphic functions in mathematics?

Meromorphic functions are important in complex analysis and number theory. They are used in the study of elliptic curves and modular forms, and are also closely related to the Riemann zeta function. Additionally, they have applications in physics and engineering, particularly in the study of dynamical systems and control theory.

Can meromorphic functions have essential singularities?

No, meromorphic functions cannot have essential singularities. This is because they are defined as a ratio of two analytic functions, and essential singularities can only occur in functions that are not analytic. Therefore, meromorphic functions can only have poles or removable singularities.

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