This solves the Schubert intersection problem on $Gr(p,m+p)$ of the $p$-planes that meet $m\cdot p$ general $m$-planes. It is a simple Schubert problem $l_1,\dotsc,l_{mp}$, where each Schubert condition $l_i$ has codimension 1. This is less general than solveSimpleSchubert, which allows up to two of the $\ell_i$ to not have codimension 1
The example below computes the two lines that meet four given lines in projective 3-space.
To verify the first intersection condition, we concatenate the first input plane with the first output plane. The determinant of the concatenated matrix should have real and imaginary parts of the same magnitude as the machine precision.
i1 : (ipt, otp) := PieriHomotopies(2,2);
|
i2 : print ipt
{| -.5 .361294+.0806396ii |, | -.5 .216008+.134438ii |, | -.5 .284982+.222929ii |, | -.5 .22015-.484395ii |}
| .0560812+.496845ii -.345688-.122686ii | | .425294-.262917ii -.25443-.000255548ii | | -.424034+.264943ii -.612614+.0126839ii | | -.495841+.064355ii -.407594-.169567ii |
| -.312843+.390038ii -.575119+.134847ii | | .378729+.326442ii -.510118+.406777ii | | .356538-.350544ii -.489129+.356105ii | | -.498392-.0400682ii -.307603+.337509ii |
| -.499532+.0216369ii -.0303484+.615261ii | | -.498042+.0442031ii -.517987+.42016ii | | -.487797-.109792ii -.348876-.0765384ii | | -.0473766+.49775ii .384254-.407219ii |
|
i3 : print otp
{| 1 0 |, | 1 0 |}
| -.0995275-.303491ii .10685+.135117ii | | .122055+2.67619ii .77208+.270864ii |
| -1.00317+1.13845ii 1 | | .855829+3.01373ii 1 |
| 0 1.04563+.64657ii | | 0 -.205684-.309834ii |
|
i4 : in0 = ipt_0
o4 = | -.5 .361294+.0806396ii |
| .0560812+.496845ii -.345688-.122686ii |
| -.312843+.390038ii -.575119+.134847ii |
| -.499532+.0216369ii -.0303484+.615261ii |
4 2
o4 : Matrix CC <--- CC
53 53
|
i5 : out0 = otp_0
o5 = | 1 0 |
| -.0995275-.303491ii .10685+.135117ii |
| -1.00317+1.13845ii 1 |
| 0 1.04563+.64657ii |
4 2
o5 : Matrix CC <--- CC
53 53
|
i6 : m = in0|out0
o6 = | -.5 .361294+.0806396ii 1
| .0560812+.496845ii -.345688-.122686ii -.0995275-.303491ii
| -.312843+.390038ii -.575119+.134847ii -1.00317+1.13845ii
| -.499532+.0216369ii -.0303484+.615261ii 0
------------------------------------------------------------------------
0 |
.10685+.135117ii |
1 |
1.04563+.64657ii |
4 4
o6 : Matrix CC <--- CC
53 53
|
i7 : det m
o7 = 1.39373833114133e-15-5.34558582120369e-18*ii
o7 : CC (of precision 53)
|