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In an article I'm reading the author says that a certain set of polynomials is a basis for an ideal $I\subset \Bbb Z[x_1,\dots,x_n]$. However, I have only ever seen Groebner basis in the context of polynomial rings over a field. I also cannot find any sources discussing groebner bases over general rings, or even PIDs. Does anyone know of such a source? Or does anyone know if there are important differences between the characteristics of Groebner bases over PIDs vs Groebner bases over a field?

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  • $\begingroup$ Can't you compute the grobner basis over $\mathbb Q$ and then clear the (finitely many) denominators? $\endgroup$
    – Mark
    Dec 22, 2016 at 20:18
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    $\begingroup$ @Mark I'm not sure that works... For example if we have $I=\langle 2x \rangle\subset \Bbb Z[x]$, then in $\Bbb Q[x]$ we have that $I=\langle x\rangle$, and this is a Groebner basis. However, $x\not\in I\subset \Bbb Z[x]$ $\endgroup$ Dec 23, 2016 at 9:36

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The book Gröbner bases by Thomas Becker and Volker Weispfenning (on springer) has a chapter on Gröbner bases over other kinds of rings in the end.

But yeah, as mentioned in the comments, $\mathbb Z$ is a subring of $\mathbb Q$ so it could probably be much easier than that.

Abstract:

The material in this section is not needed for the remaining sections of this chapter. Here, we will generalize the theory of Gröbner bases to polynomial rings over principal ideal domains. We will show that for every given finite subset F of such a polynomial ring, the equivalence problem for the ideal Id(F) is solvable by means of a Gröbner basis construction. The reduction relation will not in general allow the computation of unique normal forms, but it will be such that f ∈ Id(F) iff every normal form of f equals 0. This is good enough for the solution of the equivalence problem (cf. Theorem 5.55). For Euclidean domains that allow the computation of unique remainders, we will even obtain a reduction relation with unique normal forms.

Edit:

Bernard beat me to it, but the book by Adams and Lostaunau (published by the AMS) has a chapter on it as well. That book is in general far more readable than the B&W book in my opinion! I couldn't remember the authors so I was looking through the bib file I used when I referenced it, but then Bernard reminded me. It is better recommendation I think!

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You also can take a look at chapter 4 of An Introduction to Gröbner Bases, by William W. Adams & Philippe Loustounau, Graduate Studies in Mathematics, vol. 1 (AMS Publications).

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  • $\begingroup$ Haha, was just about to edit my post to include that book... $\endgroup$
    – neptun
    Dec 22, 2016 at 20:26
  • $\begingroup$ Thank you for your suggestion too $\endgroup$ Dec 23, 2016 at 9:31

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