Finiteness of an algebraic variety $V(I) subseteq mathbb{C}^n$ where $I$ is a zero dimensional ideal of...
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How to demonstrate this?:
Let I be an ideal of $mathbb{C}[x_1,dots,x_n]$ such that $frac{mathbb{C}[x_1,dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) subseteq mathbb{C}^n$ is finite.
I have thought about using a theorem stating that $I$ is a zero dimensional ideal if and only if for each $i$ exists $g_i in G$ ($G$ is a Gröbner basis of $I$) and $beta_i in mathbb{N}$ such that LM($g_i$)=$x_i^{beta_i}$ (LM refers to leading monomial).
complex-numbers ideals groebner-basis
asked Jan 15 at 10:45
math94math94
62
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add a comment |
$begingroup$
How to demonstrate this?:
Let I be an ideal of $mathbb{C}[x_1,dots,x_n]$ such that $frac{mathbb{C}[x_1,dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) subseteq mathbb{C}^n$ is finite.
I have thought about using a theorem stating that $I$ is a zero dimensional ideal if and only if for each $i$ exists $g_i in G$ ($G$ is a Gröbner basis of $I$) and $beta_i in mathbb{N}$ such that LM($g_i$)=$x_i^{beta_i}$ (LM refers to leading monomial).
complex-numbers ideals groebner-basis
asked Jan 15 at 10:45
math94math94
62
$endgroup$
$begingroup$
I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12
add a comment |
$begingroup$
How to demonstrate this?:
Let I be an ideal of $mathbb{C}[x_1,dots,x_n]$ such that $frac{mathbb{C}[x_1,dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) subseteq mathbb{C}^n$ is finite.
I have thought about using a theorem stating that $I$ is a zero dimensional ideal if and only if for each $i$ exists $g_i in G$ ($G$ is a Gröbner basis of $I$) and $beta_i in mathbb{N}$ such that LM($g_i$)=$x_i^{beta_i}$ (LM refers to leading monomial).
complex-numbers ideals groebner-basis
asked Jan 15 at 10:45
math94math94
62
$endgroup$
How to demonstrate this?:
Let I be an ideal of $mathbb{C}[x_1,dots,x_n]$ such that $frac{mathbb{C}[x_1,dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) subseteq mathbb{C}^n$ is finite.
I have thought about using a theorem stating that $I$ is a zero dimensional ideal if and only if for each $i$ exists $g_i in G$ ($G$ is a Gröbner basis of $I$) and $beta_i in mathbb{N}$ such that LM($g_i$)=$x_i^{beta_i}$ (LM refers to leading monomial).
complex-numbers ideals groebner-basis
complex-numbers ideals groebner-basis
asked Jan 15 at 10:45
math94math94
62
asked Jan 15 at 10:45
math94math94
62
asked Jan 15 at 10:45
math94math94
62
asked Jan 15 at 10:45
math94math94
62
asked Jan 15 at 10:45
math94math94
62
62
$begingroup$
I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12
add a comment |
$begingroup$
I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12
$begingroup$
I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12
$begingroup$
I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12
add a comment |
1 Answer
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Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
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1 Answer
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1 Answer
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$begingroup$
Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
$endgroup$
add a comment |
$begingroup$
Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
$endgroup$
add a comment |
$begingroup$
Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
$endgroup$
Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 11:44
WuestenfuxWuestenfux
5,5911513
5,5911513
add a comment |
add a comment |
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I just want to demonstrate the theoretical statement and I have no examples.
$endgroup$
– math94
Jan 16 at 13:12