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Math Help - Primary decomposition

  1. #1
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    Primary decomposition

    Let P_1=<x,y>, P_2=<x,z>, P_3=<y,z> \in \mathbb{Q}[x,y,z].
    Compute an irredundant primary decomposition and the associated primes of the product P_1P_2P_3.

    How can I compute a primary decomposition???
    If you serve solution of this problem, then it's very helpful for me.
    Please help me..
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  2. #2
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    Quote Originally Posted by Stiger View Post
    Let P_1=<x,y>, P_2=<x,z>, P_3=<y,z> \in \mathbb{Q}[x,y,z].
    Compute an irredundant primary decomposition and the associated primes of the product P_1P_2P_3.

    How can I compute a primary decomposition???
    If you serve solution of this problem, then it's very helpful for me.
    Please help me..
    an irredundant primary decomposition is: <x,y> \cap <x,z> \cap <y,z> \cap <x^2,y^2,z^2,xyz> and the associated primes are obviously <x,y>, <x,z>, <y,z>, <x,y,z>. the idea is

    to first write your ideal as an ideal generated by some monomials. in here we have: P_1P_2P_3=<x^2y,x^2z,y^2x,y^2z,z^2x,z^2y,xyz>. this is equal to the intersection of all ideals generated by

    7 monomials, each chosen from one of the generators of P_1P_2P_3. many of these ideals are subsets of some others and so we just delete them. it is important to know that a monomial ideal in

    a polynomial ring k[x_1,x_2, \cdots , x_n], where k is a field, is primary if and only if it's in the form <x_{j_1}^{n_1}, x_{j_2}^{n_2}, \cdots , x_{j_r}^{n_r}, u_1, u_2, \cdots , u_s>, where u_1, u_2, \cdots , u_s are monomials in x_{j_1} , x_{j_2}, \cdots , x_{j_r}. for

    example in k[x,y,z], the ideal <x^2,y^2,z^2,xyz> is primary but <x^2,y^2, yz> is not primary.
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