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SymbolicPowers :: Monomial Curves

Monomial Curves

To test containments of symbolic and ordinary powers of ideals defining monomial curves, we can skip the step where we define the ideals.

For example, if I is the ideal defining the monomial curve defined by t3, t4, t5 over ℤ/101, we can ask whether I(3) ⊆I2:

i1 : symbolicContainmentMonomialCurve(ZZ/101,{3,4,5},3,2)

o1 = true

Or we simply ask for the symbolic powers of these ideals. For example, here is the third of the same ideal:

i2 : symbolicPowerMonomialCurve(ZZ/101,{3,4,5},3)

             6       4      2 2 2    3 3   2 5     3 3     4   2    4 2  
o2 = ideal (c  - 3b*c d + 3b c d  - b d , b c  - 2b c d + b c*d  - c d  +
     ------------------------------------------------------------------------
         2 3    2 4   3 4     4 2     5 2    5        3 2    2   3   5 2  
     2b*c d  - b d , b c  - 2b c d + b d  - c d + 2b*c d  - b c*d , b c  -
     ------------------------------------------------------------------------
      6       5     2 3      3   2    2 3      4   7     4 2      5 2    5   
     b d + b*c  - 4b c d + 3b c*d  + c d  - b*d , b c - b c d - 2b d  + c d -
     ------------------------------------------------------------------------
         3 2     2   3    5   8    4 3     5         4      2 2 2    3 3  
     3b*c d  + 5b c*d  - d , b  + b c  - 4b c*d - b*c d + 3b c d  + b d  -
     ------------------------------------------------------------------------
        4
     c*d )

               ZZ
o2 : Ideal of ---[b, c, d]
              101