# Software computation with arithmetic schemes

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?

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Magma can do Groebner bases over the integers, and IIRC primary decomposition as well. It will also do Groebner bases over finite truncations of p-adic rings. –  Martin Bright Aug 6 '13 at 7:27
Thanks Martin, I believe Sage also does Groebner bases over $\mathbb{Z}$ and Magma more generally over Euclidean domains. However, neither does primary decomposition or computes the radical. –  LMN Aug 6 '13 at 17:40
I find it very strange/hard to believe that there is an algorithm to compute the integral closure (which is generally considered "hard") for interesting rings ($\mathbb{Z}$-algebras above), but that there is no implementation. –  LMN Aug 6 '13 at 17:43