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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 = | 7560x_1^4+14085x_1^3x_2-6272x_1^2x_2^2+13637x_1x_2^3-13542x_2^4-7995x_
     ------------------------------------------------------------------------
     1^3x_3-626x_1^2x_2x_3-2123x_1x_2^2x_3-248x_2^3x_3+6204x_1^2x_3^2-4216x_
     ------------------------------------------------------------------------
     1x_2x_3^2-3315x_2^2x_3^2+1638x_1x_3^3-15134x_2x_3^3+7131x_3^4 |

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

o3 = | x_2^2x_3-5257x_1x_3^2-14126x_2x_3^2+13271x_3^3
     ------------------------------------------------------------------------
     x_1x_2x_3-1334x_1x_3^2-13597x_2x_3^2+15970x_3^3
     ------------------------------------------------------------------------
     x_1^2x_3+3329x_1x_3^2-13705x_2x_3^2+1806x_3^3
     ------------------------------------------------------------------------
     x_2^3-9755x_1x_3^2+1862x_2x_3^2-3811x_3^3
     ------------------------------------------------------------------------
     x_1x_2^2-10154x_1x_3^2-7490x_2x_3^2-8833x_3^3
     ------------------------------------------------------------------------
     x_1^2x_2-14177x_1x_3^2-2179x_2x_3^2+2927x_3^3
     ------------------------------------------------------------------------
     x_1^3-12337x_1x_3^2-4450x_2x_3^2+8111x_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 :