Computing Regions (in a Hyperplane Arrangment) with Zaslavsky' s Theorem

This notebook was written by Nathan Williams.  It is a very quick implementation of Zaslavasky’s Theorem.

Usage. Type ComputeRegions[a list of equations defining hyperplanes in the n variables x1,....,xn , n].

Example. The Braid arrangement in ZaslavskyRegions_1.gif has n! regions.

In[3173]:=

ZaslavskyRegions_2.gif

Out[3173]=

ZaslavskyRegions_3.gif

ZaslavskyRegions_4.gif

ZaslavskyRegions_5.gif

Example. The Linial arrangement in ZaslavskyRegions_6.gif has ZaslavskyRegions_7.gif regions.

ZaslavskyRegions_8.gif

ZaslavskyRegions_9.gif

Example. The Shi arrangement in ZaslavskyRegions_10.gif has ZaslavskyRegions_11.gif regions.

ZaslavskyRegions_12.gif

ZaslavskyRegions_13.gif

Example. The Catalan arrangement in ZaslavskyRegions_14.gif has ZaslavskyRegions_15.gif regions.

ZaslavskyRegions_16.gif

ZaslavskyRegions_17.gif

Code

In[3174]:=

ZaslavskyRegions_18.gif

Spikey Created with Wolfram Mathematica 8.0