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Math Help - Affine Algebraic Variety.

  1. #1
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    Affine Algebraic Variety.

    Find an ideal I in K[x,y] such that V(I) consists of the point (1,1) and the lines x=0 and y=x+1

    Any help would be greatly appreciated...
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  2. #2
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  3. #3
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    Quote Originally Posted by Opalg View Post
    unfortunately tonio's answer is not correct because in his example we have V(I)=\{(0,1)\}. recall that for any ideal I of K[x,y] we define V(I)=\{p \in K^2: \ f(p)=0, \ \forall f \in I \}.

    anyway, to answer the question, you can choose I=\langle x(y-x-1)(x-1), \ x(y-x-1)(y-1) \rangle. then it's easy to see that V(I) is the set you're looking for.
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  4. #4
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    Quote Originally Posted by NonCommAlg View Post
    unfortunately tonio's answer is not correct because in his example we have V(I)=\{(0,1)\}. recall that for any ideal I of K[x,y] we define V(I)=\{p \in K^2: \ f(p)=0, \ \forall f \in I \}.

    anyway, to answer the question, you can choose I=\langle x(y-x-1)(x-1), \ x(y-x-1)(y-1) \rangle. then it's easy to see that V(I) is the set you're looking for.

    Why "unfortunately"? My idea was wrong but, perhaps, it could push the OP in the right direction. Anyway, she/he now has the correct answer and can compare. Nothing lost, all is profit. Thanx for the correct answer.

    Tonio
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