TITLE: Software computation with arithmetic schemes
QUESTION [6 upvotes]: For rings such as $\mathbb{Z}[x,y]$ is there software to compute any of:
1.) The integral closure of $\mathbb{Z}[x,y]/(f)$. de Jong has a very general algorithm that works in this context (http://arxiv.org/abs/alg-geom/9704017)
2.) The radical of an ideal $I$.
3.) Minimal primes of an ideal/the primary decomposition.
4.) The dimension of $\mathbb{Z}[x,y]/I$
5.) Any invariants of a singularity on such a scheme.
Over fields $\mathbb{Q}, \mathbb{F}_p$ I think Sage, Macaulay 2 etc., implement 1-4. Are 1-4 difficult to compute for such rings, or are there just no implementations yet?

REPLY [2 votes]: To my knowledge, 
2-4 are also implemented in Singular (http://www.singular.uni-kl.de/), in primdecint.lib or in kernel, but the Gröbner basis computation over integers and other algrorithms in Singular seems still buggy.
We are currently trying to pinpoint the bugs and fix them in the development version: https://github.com/Singular/Sources
Their bugtracker: www.singular.uni-kl.de:8002/trac/report/1?sort=ticket&asc=0&page=1
Radical over integers is computed in Singular via radicalZ(I), dimension via dim(std(I)) and the primary decomposition via primdecZ(I). Sagemath has an interface to Singular 3.1.6 but not to the recent version.
Jack