# An Involutive GVW Algorithm and the Computation of Pommaret Bases

## Abstract

The GVW algorithm computes simultaneously Gröbner bases of a given ideal and of the syzygy module of the given generating set. In this work, we discuss an extension of it to involutive bases. Pommaret bases play here a special role in several respects. We distinguish between a fully involutive GVW algorithm which determines involutive bases for both the given ideal and the syzygy module and a semi-involutive version which computes for the syzygy module only an ordinary Gröbner basis. A prototype implementation of the developed algorithms in Maple is described.

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## Notes

1. By this we mean that they aimed to compute an involutive basis of an ideal and a Gröbner basis of its syzygy module.

2. A regular normal form is the result of only regular reduction steps until no regular reduction is possible anymore. A regular normal form does not have to be unique as we are, in general, not reducing regular with respect to a Gröbner basis in the v-part.

3. This means that no more involutively regular reduction steps are possible.

4. This notion of a strong L-basis is not related to strong or weak involutive bases. However, it can be shown, that from a strong L-basis, two weak involutive bases will arise (see [11, Prop. 4.3.3], whose proof can easily be adapted to a general involutive division L).

5. This means, that the set of v-parts of G does not contain two elements $$v_1,v_2$$ such that $${{\,\mathrm{lt}\,}}(v_1)\mid _{L,B_v}{{\,\mathrm{lt}\,}}(v_2)$$.

6. In this work, every variable with an index smaller or equal to the class of a term $$t'$$, written cls$$(t'){:}{=}\min \{r: x_r\mid t'\}$$, is Pommaret multiplicative for $$t'$$.

7. This means, that there is no element in G that can be involtuively regular reduced by G.

8. Here, we do not want to discuss if it is even finished up to elements with signature $${{\,\mathrm{lt}\,}}(\varvec{u})$$.

9. Remember that we are only interested in a strong P-basis for $$\prec _1=\prec _\text {degrevlex}$$.

10. Remember that q is an upper bound, or the Castelnuovo–Mumford regularity.

11. Otherwise the algorithm would have computed Syz(F) and $${{\,\mathrm{lt}\,}}(\varvec{u}')$$ could not be involutively irreducible by Syz(F).

12. Here, we mean the common notion of involutive reducibility without any restrictions by a u-part.

13. This lemma is an involutive variant of a claim embedded in Theorem 3.1 from [6].

14. We will introduce it in Sect. 4.

15. Also, we will discuss later in this remark how to deal with an error message for Syz(F) coming from the Pommaret version of the algorithm.

16. For the argument corresponding to Syz(F), one may look up the arguments in the proof of correctness of the (fully) involutive GVW algorithm.

17. This is a property of the Janet division [18, p. 67].

18. This means that in its current form the implementation is not yet optimised and will have problems with larger examples. The Maple codes of our programs are available at http://www.mathematik.uni-kassel.de/~izgin/publications.php?lang=en.

19. Remember that we have an index of safety for every coordinate transformation that we perform.

20. The Maple code is available at http://www.mathematik.uni-kassel.de/~izgin/publications.php?lang=en.

## References

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Hashemi, A., Izgin, T., Robertz, D. et al. An Involutive GVW Algorithm and the Computation of Pommaret Bases. Math.Comput.Sci. 15, 419–452 (2021). https://doi.org/10.1007/s11786-021-00512-5

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• DOI: https://doi.org/10.1007/s11786-021-00512-5

### Keywords

• Gröbner bases
• Module of syzygies
• Signature-based algorithms
• The GVW algorithm
• Involutive bases
• Quasi-stable position
• Linear coordinate transformations
• Pommaret bases

• 13P10
• 68W30