Retrieving a function of many complex variables from its manifold of zeros The Next CEO of Stack Overflowquestion about a complex manifoldA complex valued continuous function which is holomorphic outside of its zerosImplicit function theorem for several complex variables.Zeros of complex function sequence (Application of Rouche's Theorem).Help needed to understand statements about toruschange of complex variablesWhat is the $frac12$ representation of $U(1)$?A limit of a complex function in several variablesUniqueness of factorization into irreducible factors of a several complex variables function.On Several Complex variables
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Retrieving a function of many complex variables from its manifold of zeros
The Next CEO of Stack Overflowquestion about a complex manifoldA complex valued continuous function which is holomorphic outside of its zerosImplicit function theorem for several complex variables.Zeros of complex function sequence (Application of Rouche's Theorem).Help needed to understand statements about toruschange of complex variablesWhat is the $frac12$ representation of $U(1)$?A limit of a complex function in several variablesUniqueness of factorization into irreducible factors of a several complex variables function.On Several Complex variables
$begingroup$
Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The states belong to a Hilbert space with the scalar product defined with the help of the measure $exp(-sum_iz_i^star z_i) d z_1... d z_N$.
Now, the question is: is it possible to retrieve a quantum state from its nodes? That is, is it possible to retrieve a holomorphic (entire) function of $N$ variables from its manifold of zeros?
As far as I understand, in the case of $N = 1$ we have the Weierstrass factorization theorem which gives a positive answer. So, does there exist a generalization of that theorem to $N > 1$?
To be even more ambitious: let us suppose we have a curve in $Bbb R^3$, a torus knot, say. Given the information that it is an intersection of a complex curve $K$ with a sphere, and the form of the knot, can we retrieve the entire function $f(z_1, z_2)$ such that $f(z_1, z_2) = 0$ is an equation of $K$?
I apologize if the question is naive, and imprecise; I'm a physicist, not a mathematician. And thank you for your attention.
algebraic-geometry mathematical-physics quantum-mechanics several-complex-variables
edited Mar 20 at 14:24
Andrews
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
$endgroup$
add a comment |
$begingroup$
Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The states belong to a Hilbert space with the scalar product defined with the help of the measure $exp(-sum_iz_i^star z_i) d z_1... d z_N$.
Now, the question is: is it possible to retrieve a quantum state from its nodes? That is, is it possible to retrieve a holomorphic (entire) function of $N$ variables from its manifold of zeros?
As far as I understand, in the case of $N = 1$ we have the Weierstrass factorization theorem which gives a positive answer. So, does there exist a generalization of that theorem to $N > 1$?
To be even more ambitious: let us suppose we have a curve in $Bbb R^3$, a torus knot, say. Given the information that it is an intersection of a complex curve $K$ with a sphere, and the form of the knot, can we retrieve the entire function $f(z_1, z_2)$ such that $f(z_1, z_2) = 0$ is an equation of $K$?
I apologize if the question is naive, and imprecise; I'm a physicist, not a mathematician. And thank you for your attention.
algebraic-geometry mathematical-physics quantum-mechanics several-complex-variables
edited Mar 20 at 14:24
Andrews
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
$endgroup$
1
$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17
add a comment |
$begingroup$
Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The states belong to a Hilbert space with the scalar product defined with the help of the measure $exp(-sum_iz_i^star z_i) d z_1... d z_N$.
Now, the question is: is it possible to retrieve a quantum state from its nodes? That is, is it possible to retrieve a holomorphic (entire) function of $N$ variables from its manifold of zeros?
As far as I understand, in the case of $N = 1$ we have the Weierstrass factorization theorem which gives a positive answer. So, does there exist a generalization of that theorem to $N > 1$?
To be even more ambitious: let us suppose we have a curve in $Bbb R^3$, a torus knot, say. Given the information that it is an intersection of a complex curve $K$ with a sphere, and the form of the knot, can we retrieve the entire function $f(z_1, z_2)$ such that $f(z_1, z_2) = 0$ is an equation of $K$?
I apologize if the question is naive, and imprecise; I'm a physicist, not a mathematician. And thank you for your attention.
algebraic-geometry mathematical-physics quantum-mechanics several-complex-variables
edited Mar 20 at 14:24
Andrews
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
$endgroup$
Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The states belong to a Hilbert space with the scalar product defined with the help of the measure $exp(-sum_iz_i^star z_i) d z_1... d z_N$.
Now, the question is: is it possible to retrieve a quantum state from its nodes? That is, is it possible to retrieve a holomorphic (entire) function of $N$ variables from its manifold of zeros?
As far as I understand, in the case of $N = 1$ we have the Weierstrass factorization theorem which gives a positive answer. So, does there exist a generalization of that theorem to $N > 1$?
To be even more ambitious: let us suppose we have a curve in $Bbb R^3$, a torus knot, say. Given the information that it is an intersection of a complex curve $K$ with a sphere, and the form of the knot, can we retrieve the entire function $f(z_1, z_2)$ such that $f(z_1, z_2) = 0$ is an equation of $K$?
I apologize if the question is naive, and imprecise; I'm a physicist, not a mathematician. And thank you for your attention.
algebraic-geometry mathematical-physics quantum-mechanics several-complex-variables
algebraic-geometry mathematical-physics quantum-mechanics several-complex-variables
edited Mar 20 at 14:24
Andrews
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
edited Mar 20 at 14:24
Andrews
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
edited Mar 20 at 14:24
Andrews
1,2812422
edited Mar 20 at 14:24
Andrews
1,2812422
edited Mar 20 at 14:24
Andrews
1,2812422
1,2812422
asked Mar 20 at 11:18
Hau HauHau Hau
11
asked Mar 20 at 11:18
Hau HauHau Hau
11
asked Mar 20 at 11:18
Hau HauHau Hau
11
11
1
$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17
add a comment |
1
$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17
1
1
$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17
add a comment |
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$begingroup$
What do you mean retrieve it from ? Because in the $N=1$ case, Weierstrass factorization doesn't give back the function, it gives a function with the correct zeroes
$endgroup$
– Max
Mar 20 at 12:21
$begingroup$
To add to Max's comment, in $N=1$, if you are given the zeros together with multiplicities, you can recover the function up to multiplication by $e^g(z)$, where $g$ is an arbitrary holomorphic function.
$endgroup$
– Wojowu
Mar 20 at 14:31
$begingroup$
Thank you very much for your valuable comments. In particular, many thanks to Wojowu for reminding me about the exponential factor. But I think the question remains valid if I replace "the" with "a". To repeat, is there a(!) generalization of the Weierstrass factorization theorem to several complex variables? And, what about getting a(!) complex curve from its knot in $bf R^3$? If you tell me that it is a simple exercise I should do on my own, I'll accept this as a legitimate answer, provided that you give me some hints. With best regards, Hau Hau
$endgroup$
– Hau Hau
Mar 21 at 19:07
$begingroup$
I would also like to thank jgon and YuiTo Cheng for their great editorial work.
$endgroup$
– Hau Hau
Mar 21 at 19:17