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Math Help - Algebraic Varieties and Ideals

  1. #1
    Super Member Bernhard's Avatar
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    Algebraic Varieties and Ideals

    In Chapter 1, Section 4 of Cox et al "Ideals, Varieties and Algorithms, Exercise 3(c) reads as follows:

    Prove the following equality of ideals in  \mathbb{Q}[x,y] :

     < 2x^2 + 3y^2 -11, x^2 - y^2 - 3> \ =  \ <x^2 - 4, y^2 - 1>

    Any help with this problem would be appreciated.

    Peter
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    Re: Algebraic Varieties and Ideals

    I will give you one half:

    x^2 - 4 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) + \dfrac{3}{5}(x^2 - y^2 - 3)

    y^2 - 1 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) - \dfrac{2}{5}(x^2 - y^2 - 3)

    Your turn.
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Algebraic Varieties and Ideals

    Quote Originally Posted by Deveno View Post
    I will give you one half:

    x^2 - 4 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) + \dfrac{3}{5}(x^2 - y^2 - 3)

    y^2 - 1 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) - \dfrac{2}{5}(x^2 - y^2 - 3)

    Your turn.
    Thanks for the guidance and help Deveno

    OK so we want to show that in  \mathbb{Q}[x,y] we have  <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ =  \ <x^2 - 4, y^2 - 1>


    First show that   <x^2 - 4, y^2 - 1> \ \subseteq \ <2x^2 + 3y^2 - 11, x^2 - y^2 - 3>


    From Deveno's equations above it follows that  x^2 - 4, y^2 - 1 \in  \  <2x^2 + 3y^2 - 11, x^2 - y^2 - 3>

    Thus,   <x^2 - 4, y^2 - 1> \ \subseteq \ <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> ... ... ... (1)


    Now show  <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ \subseteq \ <x^2 - 4, y^2 - 1>


    We can write  2x^2 + 3y^2 - 11 \ = \ 2(x^2 - 4) + 3(y^2 - 1)

    and  x^2 - y^2 - 3 \ = \ (x^2 - 4) - (y^2 - 1)

    Thus  2x^2 + 3y^2 - 11, x^2 - y^2 - 3 \in  <x^2 - 4, y^2 - 1>

    Thus  <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ \subseteq \ <x^2 - 4, y^2 - 1> ... ... ... (2)


    Equations (1) (2)  \ \Longrightarrow \ \mathbb{Q}[x,y] we have  <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ = \ <x^2 - 4, y^2 - 1>
    Last edited by Bernhard; August 2nd 2013 at 04:08 PM.
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  4. #4
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    Re: Algebraic Varieties and Ideals

    Note this shows the generators of ideals in Q[x,y] aren't even unique up to units. Also, this is just high-school algebra in fancier clothes.
    Thanks from Bernhard
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