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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 = | 4028x_1^4-12805x_1^3x_2+2475x_1^2x_2^2-15333x_1x_2^3-8850x_2^4-5657x_1
     ------------------------------------------------------------------------
     ^3x_3-9731x_1^2x_2x_3-11652x_1x_2^2x_3+5709x_2^3x_3-4943x_1^2x_3^2-1799x
     ------------------------------------------------------------------------
     _1x_2x_3^2+14366x_2^2x_3^2+8046x_1x_3^3-12352x_2x_3^3+5553x_3^4 |

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

o3 = | x_2^2x_3+10253x_1x_3^2+7132x_2x_3^2-9443x_3^3
     ------------------------------------------------------------------------
     x_1x_2x_3-13476x_1x_3^2+490x_2x_3^2-4298x_3^3
     ------------------------------------------------------------------------
     x_1^2x_3-13098x_1x_3^2-4986x_2x_3^2-14248x_3^3
     ------------------------------------------------------------------------
     x_2^3-2272x_1x_3^2-7000x_2x_3^2-13392x_3^3
     ------------------------------------------------------------------------
     x_1x_2^2-11056x_1x_3^2-11351x_2x_3^2-7994x_3^3
     ------------------------------------------------------------------------
     x_1^2x_2-13918x_1x_3^2-12107x_2x_3^2-11932x_3^3
     ------------------------------------------------------------------------
     x_1^3+2515x_1x_3^2+1968x_2x_3^2-32x_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 :