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fromDual -- ideal from inverse system

Synopsis

Description

For other examples, and a more precise definition, see inverse systems.
i1 : R = ZZ/32003[x_1..x_3];
i2 : g = random(R^1, R^{-4})

o2 = | 8110x_1^4-3561x_1^3x_2-6917x_1^2x_2^2+6905x_1x_2^3+10322x_2^4+1731x_1^
     ------------------------------------------------------------------------
     3x_3+1686x_1^2x_2x_3-14427x_1x_2^2x_3+10867x_2^3x_3-1199x_1^2x_3^2+
     ------------------------------------------------------------------------
     12140x_1x_2x_3^2+12150x_2^2x_3^2-9689x_1x_3^3-14379x_2x_3^3+8904x_3^4 |

             1       1
o2 : Matrix R  <--- R
i3 : f = fromDual g

o3 = | x_2^2x_3+5022x_1x_3^2+7971x_2x_3^2+5817x_3^3
     ------------------------------------------------------------------------
     x_1x_2x_3-13505x_1x_3^2-6478x_2x_3^2+14770x_3^3
     ------------------------------------------------------------------------
     x_1^2x_3-373x_1x_3^2+9449x_2x_3^2+5742x_3^3
     ------------------------------------------------------------------------
     x_2^3+4144x_1x_3^2-1337x_2x_3^2-10468x_3^3
     ------------------------------------------------------------------------
     x_1x_2^2-7405x_1x_3^2+8229x_2x_3^2-15984x_3^3
     ------------------------------------------------------------------------
     x_1^2x_2+15214x_1x_3^2-14201x_2x_3^2+5846x_3^3
     ------------------------------------------------------------------------
     x_1^3+5007x_1x_3^2+5848x_2x_3^2+9242x_3^3 |

             1       7
o3 : Matrix R  <--- R
i4 : res ideal f

      1      7      7      1
o4 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o4 : ChainComplex
i5 : betti oo

            0 1 2 3
o5 = total: 1 7 7 1
         0: 1 . . .
         1: . . . .
         2: . 7 7 .
         3: . . . .
         4: . . . 1

o5 : BettiTally

See also

Ways to use fromDual :