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Toward involutive bases over effective rings

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Abstract

In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases.

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Notes

  1. id est when, with the notation above, for each \(i,j, j> i, c_{ji}=1\) so that \(X_jX_i=X_iX_j-d_{ij}\).

  2. Which implies, but is not equivalent to, \(\mathbf{M}(h)=\mathbf{M}(f)*v\)

  3. It is well-known that the property that any monomial has at most one divisor was already shared by the original Buchberger Algorithm which reduced each term by the oldest element inclosed in the basis and thus with lower index, thus giving a unique decomposition of \(\mathbf{T}(F), F=\{g_1,\ldots ,g_u\}\) as

    $$\begin{aligned} \mathbf{T}(F)=\sqcup _{i=1}^u \mathbf{T}(g_i)\mathsf{T_i}, \text{ with } \mathsf{T_i}=\left\{ t :\, \mathbf{T}(g_i)\circ t\notin {\mathbb {I}}(\mathbf{T}(g_1),\ldots ,\mathbf{T}(g_{i-1}))\right\} . \end{aligned}$$

    A direct application of this property still within Buchberger theory has been at the center of the notion of staggered bases [17] and marked bases [3,4,5,6,7, 14]; the effort of giving a general formulation of Buchberger and Janet theory under the name of reduction structures [11], unfortunately presents a fatal flow.

    At the same time a similar decompostion was at the center of the notion of Janet-like division [20,21,22]; we propose here a decomposition in a similar framework, pointing to another potential open problem: to extend the theory of Janet-like decompositions and to produce a completion procedure in our setting adapting that stated for Janet-like division. Here we will give in Example 4 an involutive completion of the set U.

  4. id est the monoial \(\tau \in U\) such that \(M_J(\tau ,U)= \{X_1,\dots ,X_n\}\).

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Authors and Affiliations

  1. Department of Computer Science, University of Milan, Via Celoria 18, 20133, Milano, Italy

    Michela Ceria

  2. Department of Mathematics, University of Genoa, Via Dodecaneso 35, Genova, Italy

    Teo Mora

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  1. Michela Ceria
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  2. Teo Mora
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Correspondence to Michela Ceria.

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This research has been partially funded by GNSAGA—Istituto Nazionale di Alta Matematica “Francesco Severi”. The first author is thankful to this institution for its support.

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Ceria, M., Mora, T. Toward involutive bases over effective rings. AAECC 31, 359–387 (2020). https://doi.org/10.1007/s00200-020-00448-6

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