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It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner bases for ideals in $\mathbb Z[x]$. But I am having trouble defining Groebner Bases for submodule of free $\mathbb Z[x]$-modules and showing Groebner bases exist.

So my question is how do we define Groebner bases for $\mathbb Z[x]$-modules and are we able to find it algorithmically?

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  • $\begingroup$ Neither Maple nor Mathematica have that algorithm. $\endgroup$
    – user64494
    Commented Oct 3, 2019 at 18:24

1 Answer 1

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There is a description of the appropriate Groebner basis algorithm in this book:

Franz Pauer, Andreas Unterkircher. Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations. AAECC 9, 271–291 (1999)

I've implemented it in the single-variable case (in the software Regina) and I've been meaning to implement it in the multi-variable case as well. But I usually get too sad to finish – when I look at how inefficient the algorithm is. Some day I'll have it fully implemented in Regina.

The book is quite well written. I find it easy to read.

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