Finding cancelling polynomials of a set
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Let $V={(t,t^2,t^3),t in mathbb C}$. Find $I(V)$.
I found some polynomials that cancels in $V$. For example $X-Y^2$, $Y^2-Z^3$ or $X+Y^2-Z^3$ but I don't know how to find all polynomials, that is how to determine $I(V)$.
Thank you!
algebraic-geometry polynomials
asked Nov 17 at 17:21
mip
344
add a comment |
up vote
0
down vote
favorite
Let $V={(t,t^2,t^3),t in mathbb C}$. Find $I(V)$.
I found some polynomials that cancels in $V$. For example $X-Y^2$, $Y^2-Z^3$ or $X+Y^2-Z^3$ but I don't know how to find all polynomials, that is how to determine $I(V)$.
Thank you!
algebraic-geometry polynomials
asked Nov 17 at 17:21
mip
344
The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $V={(t,t^2,t^3),t in mathbb C}$. Find $I(V)$.
I found some polynomials that cancels in $V$. For example $X-Y^2$, $Y^2-Z^3$ or $X+Y^2-Z^3$ but I don't know how to find all polynomials, that is how to determine $I(V)$.
Thank you!
algebraic-geometry polynomials
asked Nov 17 at 17:21
mip
344
Let $V={(t,t^2,t^3),t in mathbb C}$. Find $I(V)$.
I found some polynomials that cancels in $V$. For example $X-Y^2$, $Y^2-Z^3$ or $X+Y^2-Z^3$ but I don't know how to find all polynomials, that is how to determine $I(V)$.
Thank you!
algebraic-geometry polynomials
algebraic-geometry polynomials
asked Nov 17 at 17:21
mip
344
asked Nov 17 at 17:21
mip
344
asked Nov 17 at 17:21
mip
344
asked Nov 17 at 17:21
mip
344
asked Nov 17 at 17:21
mip
344
344
The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23
add a comment |
The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23
The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23
add a comment |
1 Answer
1
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1
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Consider the ideal $I=langle x-t, y-t^2, z-t^3rangle$ in ${Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = Icap {Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.
I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.
answered Nov 17 at 17:31
Wuestenfux
2,6621410
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Consider the ideal $I=langle x-t, y-t^2, z-t^3rangle$ in ${Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = Icap {Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.
I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.
answered Nov 17 at 17:31
Wuestenfux
2,6621410
add a comment |
up vote
1
down vote
accepted
Consider the ideal $I=langle x-t, y-t^2, z-t^3rangle$ in ${Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = Icap {Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.
I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.
answered Nov 17 at 17:31
Wuestenfux
2,6621410
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Consider the ideal $I=langle x-t, y-t^2, z-t^3rangle$ in ${Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = Icap {Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.
I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.
answered Nov 17 at 17:31
Wuestenfux
2,6621410
Consider the ideal $I=langle x-t, y-t^2, z-t^3rangle$ in ${Bbb K}[x,y,z,t]$ with elimination order (lex) such that $t>x>y>z$. The first elimination ideal $I_1 = Icap {Bbb K}[x,y,z]$ is what you are looking for. You just need to construct the Groebner basis of $I$ and then take those generators that are polynomials in $x,y,z$. Done.
I'd consider the book of Cox et al., "Ideal, Varieties and Algorithms", Springer. Keyword: implicitization.
answered Nov 17 at 17:31
Wuestenfux
2,6621410
edited Nov 18 at 8:34
answered Nov 17 at 17:31
Wuestenfux
2,6621410
answered Nov 17 at 17:31
Wuestenfux
2,6621410
answered Nov 17 at 17:31
Wuestenfux
2,6621410
2,6621410
add a comment |
add a comment |
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The technique is to use Groebner bases.
– Wuestenfux
Nov 17 at 17:23
I'm sorry, can you please give me a link?
– mip
Nov 17 at 17:23