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MixedMultiplicity :: multiReesIdeal

multiReesIdeal -- Compute the defining ideal of multi-Rees algebra of ideals

Synopsis

Description

The function computes the defining ideal of the multi-Rees algebra of a set of ideals over a polynomial ring by computing the saturation of a binomial ideal with respect to a polynomial. The technique is a generalization of a result of D. Cox, K.-i. Lin and G. Sosa for monomial ideals over a polynomial ring.

i1 : S = QQ[x_0..x_3]

o1 = S

o1 : PolynomialRing
i2 : C = trim monomialCurveIdeal(S,{2,3,5})

                          3    2     3      2
o2 = ideal (x x  - x x , x  - x x , x  - x x )
             1 2    0 3   2    1 3   1    0 2

o2 : Ideal of S
i3 : multiReesIdeal {C}

             2                   2                    3      2          3  
o3 = ideal (x X  - x X  - x X , x X  - x X  - x X , (x  - x x )X  + (- x  +
             2 1    1 2    3 3   1 1    0 2    2 3    1    0 2  2       2  
     ------------------------------------------------------------------------
      2
     x x )X )
      1 3  3

o3 : Ideal of S[X ..X ]
                 1   3
i4 : multiReesIdeal {C,C}

                                                    2                   2    
o4 = ideal (X X  - X X , X X  - X X , X X  - X X , x X  - x X  - x X , x X  -
             3 5    2 6   3 4    1 6   2 4    1 5   2 4    1 5    3 6   1 4  
     ------------------------------------------------------------------------
                   2                   2                    3      2        
     x X  - x X , x X  - x X  - x X , x X  - x X  - x X , (x  - x x )X  + (-
      0 5    2 6   2 1    1 2    3 3   1 1    0 2    2 3    1    0 2  5     
     ------------------------------------------------------------------------
      3    2         3      2          3    2
     x  + x x )X , (x  - x x )X  + (- x  + x x )X )
      2    1 3  6    1    0 2  2       2    1 3  3

o4 : Ideal of S[X ..X ]
                 1   6

Ways to use multiReesIdeal :

For the programmer

The object multiReesIdeal is a method function with options.