Some mathematical software that I use:
excellent computational commutative algebra package. Here's
a short example of some code
I have written for use in Macaulay 2: the procedure calculates
the largest monomial ideal sitting inside a given ideal.
state of the art for polyhedral computations.
for computing the Hilbert bases of cones.
4ti2: very nice package to compute
the circuits of a given matrix, also for generating sets and Gröbner
bases of toric ideals. Also computes the marginal matrix from a specified hierarchical model!
for computing and walking around the toric Gröbner fan.
same as CaTS except for any Gröbner fan, not just toric ones. Is
also useful for tropical computations but I know less about this program
than the others above.
Practically efficient package for enumerating lattice points
in a polytope. Again, my knowledge of this is less than
the others above.
Finally, you may be interested in ubuntu
as a user-friendly, linux/debian based alternative to windows. There are
many good reasons to switch but why take it from me when you can get it
from the horse's mouth: Ctrl-Alt-Deceit(1) and
Harper's, April 2007, without