Intersection of (n-d)*d random real (n-d-1)-planes in complex n-space.

Root Counts :

--------------------------------
  n\d   2     3    4    5  6  7
--------------------------------
   4    2     1
   5    5     5      
   6   14    42   14
   7   42   462        42
   8  132  6006
   9  429
--------------------------------

Timings on SunOS mars 5.6 Generic sun4u sparc SUNW,Ultra-1 :

  Set-up : syzygies, bracket expansions, lifting to set up SAGBI homotopy
  Volume : dynamic lifting for triangulation and volume computation
  Polcon : polyhedral continuation to solve complex random system
  Fladef : flat deformation to solve random complex problem instance
  Cheat  : cheater's homotopy from random complex to random float
  Total  : total cpu time for solving the problem

-----------------------------------------------------------------------------
  d=2  :  n = 4   n = 5     n = 6     n = 7        n = 8          n = 9
----------------------------------------------------------------------------
#vars  :    4       6         8        10           12             14
#sols  :    2       5        14        42          132            429
-----------------------------------------------------------------------------
Set-up :   30ms  0s 60ms   0s170ms  0m 0s400ms   0m 0s790ms   0h 0m 1s580ms
Volume :    0ms  0s 10ms   0s 40ms  0m 0s190ms   0m 1s310ms   0h 0m 8s890ms
Polcon :   70ms  1s210ms   6s790ms  1m27s 10ms  10m16s880ms   1h15m 9s870ms
Fladef :   60ms  1s290ms   6s820ms  1m 5s990ms   7m 5s120ms   1h 1m38s280ms
Cheat  :  640ms  3s 90ms  26s740ms  2m33s570ms  22m24s270ms   2h14m33s590ms
Total  :  800ms  5s660ms  40s570ms  5m 7s190ms  39m48s420ms   4h31m32s290ms
-----------------------------------------------------------------------------
  d=3  :  n = 4   n = 5     n = 6       n = 7        n = 8  
-----------------------------------------------------------------------------
#vars  :    3       6         9          12           15
#sols  :    1       5        42         462         6006
-----------------------------------------------------------------------------
Set-up :   30ms  0s140ms  0m 0s440ms
Volume :    0ms  0s  0ms  0m 0s170ms
Polcon :   20ms  0s600ms  1m 2s750ms
Fladef :   10ms  0s540ms  0m50s590ms
Cheat  :   30ms  1s830ms  2m47s910ms
Total  :   90ms  3s120ms  4m41s910ms
-----------------------------------------------------------------------------

Observations:
  1) setting up the equations and computing the volume is minor
  2) cheater's homotopy is most time-consuming
  3) the complexity of the problem increases rapidly as n and d grow