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NumericalGodeaux :: fromPointInP3xP5ToPointInP3xP3xP3xP3

fromPointInP3xP5ToPointInP3xP3xP3xP3 -- compute a point in the model in P3xP3xP3xP3

Synopsis

Description

Since the map from the model in P3xP3xP3xP3 3 ×ℙ3 ×ℙ3 ×ℙ3 to the model in 3 ×ℙ5 has 1-dimensional rational fibers it is possible to compute a preimage point.

i1 : kk=ZZ/(nextPrime 10^3)

o1 = kk

o1 : QuotientRing
i2 : H = precomputedModelInP3xP5(kk);

                                                        1                                                  1
o2 : Matrix (kk[w , w , w , w , z , z , z , z , z , z ])  <--- (kk[w , w , w , w , z , z , z , z , z , z ])
                 0   1   2   3   0   1   2   3   4   5              0   1   2   3   0   1   2   3   4   5
i3 : pt=findPointInP3xP5(kk)

o3 = | 107 -308 280 -176 115 -298 72 378 275 252 |

              1        10
o3 : Matrix kk  <--- kk
i4 : substitute(H,pt)

o4 = 0

              1        1
o4 : Matrix kk  <--- kk
i5 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt)

o5 = | -402 -460 -378 1 -149 488 241 1 -150 4 430 1 -121 254 136 1 |

              1        16
o5 : Matrix kk  <--- kk
i6 : I=precomputedModelInP3xP3xP3xP3(kk);

o6 : Ideal of kk[b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   ]
                  0,0   0,1   0,2   0,3   1,0   1,1   1,2   1,3   2,0   2,1   2,2   2,3   3,0   3,1   3,2   3,3
i7 : sub(I,pt1)

o7 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     ------------------------------------------------------------------------
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     ------------------------------------------------------------------------
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     ------------------------------------------------------------------------
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     ------------------------------------------------------------------------
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

o7 : Ideal of kk

Using pointOnARationalCodim1Hypersurface it is also possible to find points over .

i8 : kk=QQ

o8 = QQ

o8 : Ring
i9 : H = precomputedModelInP3xP5(QQ);

                                                        1                                                  1
o9 : Matrix (QQ[w , w , w , w , z , z , z , z , z , z ])  <--- (QQ[w , w , w , w , z , z , z , z , z , z ])
                 0   1   2   3   0   1   2   3   4   5              0   1   2   3   0   1   2   3   4   5
i10 : pt=pointOnARationalCodim1Hypersurface(100)

o10 = | 41 22 43 -325434931159/10565274372 -48060/559 29 22 27 13 35 |

               1        10
o10 : Matrix QQ  <--- QQ
i11 : substitute(H,pt)

o11 = 0

               1        1
o11 : Matrix QQ  <--- QQ
i12 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt)

o12 = | -150933551055440670543260 -104853535794548305497360
      -----------------------------------------------------------------------
      -2744060229829237021209 42685381352899242552140 17511780422500
      -----------------------------------------------------------------------
      -19187086745868 -14555720979624 1488653282007 12080479342733119509776
      -----------------------------------------------------------------------
      -5242676789727415274868 -2524251787646533280492 436889732477284606239
      -----------------------------------------------------------------------
      -433176249252 -232436036184 -454306797996 325434931159 |

               1        16
o12 : Matrix QQ  <--- QQ
i13 : I=precomputedModelInP3xP3xP3xP3(kk);

o13 : Ideal of QQ[b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   , b   ]
                   0,0   0,1   0,2   0,3   1,0   1,1   1,2   1,3   2,0   2,1   2,2   2,3   3,0   3,1   3,2   3,3
i14 : sub(I,pt1)

o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      -----------------------------------------------------------------------
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      -----------------------------------------------------------------------
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      -----------------------------------------------------------------------
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      -----------------------------------------------------------------------
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

o14 : Ideal of QQ

See also

Ways to use fromPointInP3xP5ToPointInP3xP3xP3xP3 :