Intersection of (n-d)*d random real (n-d-1)-planes in complex n-space.
Root Counts :
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n\d 2 3 4 5 6 7
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4 2 1
5 5 5
6 14 42 14
7 42 462 42
8 132 6006
9 429
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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
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d=2 : n = 4 n = 5 n = 6 n = 7 n = 8 n = 9
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#vars : 4 6 8 10 12 14
#sols : 2 5 14 42 132 429
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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
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d=3 : n = 4 n = 5 n = 6 n = 7 n = 8
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#vars : 3 6 9 12 15
#sols : 1 5 42 462 6006
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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
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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