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# randomExtension -- a random extension of a vector bundle on E by another vector bundle

- Usage:
    `V=randomExtension(V1,V2)`
    `V=randomExtension(Y1,Y2)`
- Inputs:
  - V1, an instance of the type VectorBundleOnE,
  - V2, an instance of the type VectorBundleOnE,
  - Y1, a matrix, a matrix which is the yAction of some VectorBundleOnE
  - Y2, a matrix, a matrix which is the yAction of some VectorBundleOnE
- Outputs:
  - V, an instance of the type VectorBundleOnE,

## Description

Chooses a random extension of V2 by V1, where V1, V2 are vector bundles on E represented by the type VectorBundleOnE.

    i1 : setRandomSeed 0
     -- setting random seed to 0

    o1 = 0

    i2 : kk=ZZ/101;

    i3 : g=1;

    i4 : rNP=randNicePencil(kk,g);

    i5 : cM=cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing);

    i6 : f=cM.hyperellipticBranchEquation

              3       2 2        3      4
    o6 = - 12s t - 50s t  - 16s*t  + 47t

    o6 : kk[s, t]

    i7 : L1=randomLineBundle(0,f)

    o7 = VectorBundleOnE{...1...}

    o7 : VectorBundleOnE

    i8 : L2=randomLineBundle(2,f)

    o8 = VectorBundleOnE{...1...}

    o8 : VectorBundleOnE

    i9 : V=randomExtension(L1,L2)

    o9 = VectorBundleOnE{...1...}

    o9 : VectorBundleOnE

    i10 : V.yAction

    o10 = {-1} | -18s2-13st-43t2 13s2+45st-16t2 40s4+37s3t+16s2t2+35st3
          {-1} | 45s2-14st-t2    18s2+13st+43t2 11s4+45s3t+6s2t2-5st3+3t4
          {1}  | 0               0              48s2+36st+35t2
          {-1} | 0               0              44s4+s3t-22s2t2+33st3-t4
          -----------------------------------------------------------------------
          -42s2-33st-48t2 |
          39s2-30st+32t2  |
          -34             |
          -48s2-36st-35t2 |

                           4               4
    o10 : Matrix (kk[s, t])  <-- (kk[s, t])

    i11 : degOnE V

    o11 = 2

    i12 : V1=randomExtension(L2,V)

    o12 = VectorBundleOnE{...1...}

    o12 : VectorBundleOnE

    i13 : V1.yAction

    o13 = {1}  | 48s2+36st+35t2           -34             36
          {-1} | 44s4+s3t-22s2t2+33st3-t4 -48s2-36st-35t2 29s2+40st-43t2
          {-1} | 0                        0               -18s2-13st-43t2
          {-1} | 0                        0               45s2-14st-t2
          {1}  | 0                        0               0
          {-1} | 0                        0               0
          -----------------------------------------------------------------------
          -38            -22s2-40st+6t2               -29             |
          41s2-19st-15t2 -28s4-20s3t+11s2t2+31st3-5t4 29s2-3st+50t2   |
          13s2+45st-16t2 40s4+37s3t+16s2t2+35st3      -42s2-33st-48t2 |
          18s2+13st+43t2 11s4+45s3t+6s2t2-5st3+3t4    39s2-30st+32t2  |
          0              48s2+36st+35t2               -34             |
          0              44s4+s3t-22s2t2+33st3-t4     -48s2-36st-35t2 |

                           6               6
    o13 : Matrix (kk[s, t])  <-- (kk[s, t])

    i14 : degOnE V1

    o14 = 4

## See also

- randomLineBundle -- a random balanced line bundle on the hyperelliptic curve
- vectorBundleOnE -- creates a VectorBundleOnE, represented as a matrix factorization
- VectorBundleOnE -- vector bundle on a hyperelliptic curve E
- degOnE -- degree of a vector bundle on E

## Ways to use randomExtension:

- `randomExtension(Matrix,Matrix)`
- `randomExtension(VectorBundleOnE,VectorBundleOnE)`

## For the programmer

The object randomExtension is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:3195:0.