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







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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















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







share|cite|improve this question






















  • 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













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







share|cite|improve this question













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






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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


















  • 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










1 Answer
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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.






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    1 Answer
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    active

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    up vote
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    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.






    share|cite|improve this answer



























      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.






      share|cite|improve this answer

























        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.






        share|cite|improve this answer














        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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 18 at 8:34

























        answered Nov 17 at 17:31









        Wuestenfux

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