## 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

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

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.

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

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*).This means, that the set of v-parts of

*G*does not contain two elements \(v_1,v_2\) such that \({{\,\mathrm{lt}\,}}(v_1)\mid _{L,B_v}{{\,\mathrm{lt}\,}}(v_2)\).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 \(t'\).This means, that there is no element in

*G*that can be involtuively regular reduced by*G*.Here, we do not want to discuss if it is even finished up to elements with signature \({{\,\mathrm{lt}\,}}(\varvec{u})\).

Remember that we are only interested in a strong

*P*-basis for \(\prec _1=\prec _\text {degrevlex}\).Remember that

*q*is an upper bound, or the Castelnuovo–Mumford regularity.Otherwise the algorithm would have computed

*Syz*(*F*) and \({{\,\mathrm{lt}\,}}(\varvec{u}')\) could not be involutively irreducible by*Syz*(*F*).Here, we mean the common notion of involutive reducibility without any restrictions by a u-part.

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

We will introduce it in Sect. 4.

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.For the argument corresponding to

*Syz*(*F*), one may look up the arguments in the proof of correctness of the (fully) involutive GVW algorithm.This is a property of the Janet division [18, p. 67].

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.

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

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

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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

### Mathematics Subject Classification

- 13P10
- 68W30