Mixing
Algebraic dependence of meromorphic functions on a compact Riemann surface
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I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P in mathbb{C}[X,Y]$ such that $P(f,g) = 0$.
I have seen this question Meromorphic Function in a Compact Riemann Surface however I didn't find the discussion very useful.
Firstly, why should I expect such a result to be true?
Also, how can I go about proving this?
One thing I noticed is a polynomial in $f,g$ can be written as $(1,f,...,f^r)A(1,g,...,g^r)^t$ for some matrix of coefficients $A$, but I did not find a way to use this.
complex-analysis riemann-surfaces
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
$endgroup$
add a comment |
$begingroup$
I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P in mathbb{C}[X,Y]$ such that $P(f,g) = 0$.
I have seen this question Meromorphic Function in a Compact Riemann Surface however I didn't find the discussion very useful.
Firstly, why should I expect such a result to be true?
Also, how can I go about proving this?
One thing I noticed is a polynomial in $f,g$ can be written as $(1,f,...,f^r)A(1,g,...,g^r)^t$ for some matrix of coefficients $A$, but I did not find a way to use this.
complex-analysis riemann-surfaces
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
$endgroup$
$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01
add a comment |
$begingroup$
I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P in mathbb{C}[X,Y]$ such that $P(f,g) = 0$.
I have seen this question Meromorphic Function in a Compact Riemann Surface however I didn't find the discussion very useful.
Firstly, why should I expect such a result to be true?
Also, how can I go about proving this?
One thing I noticed is a polynomial in $f,g$ can be written as $(1,f,...,f^r)A(1,g,...,g^r)^t$ for some matrix of coefficients $A$, but I did not find a way to use this.
complex-analysis riemann-surfaces
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
$endgroup$
I am trying to show given two meromorphic functions $f,g$ on a compact Riemann surface $X$ that there exists a non-zero polynomial $P in mathbb{C}[X,Y]$ such that $P(f,g) = 0$.
I have seen this question Meromorphic Function in a Compact Riemann Surface however I didn't find the discussion very useful.
Firstly, why should I expect such a result to be true?
Also, how can I go about proving this?
One thing I noticed is a polynomial in $f,g$ can be written as $(1,f,...,f^r)A(1,g,...,g^r)^t$ for some matrix of coefficients $A$, but I did not find a way to use this.
complex-analysis riemann-surfaces
complex-analysis riemann-surfaces
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
asked Jan 10 at 19:41
Evgeny TEvgeny T
709414
709414
$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01
add a comment |
$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01
$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01
$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01
add a comment |
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$begingroup$
If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces".
$endgroup$
– Moishe Cohen
Jan 12 at 17:01