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Degröbnerization: a political manifesto

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics.

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Notes

  1. Note that we always take \(b_1=1\); the algorithms could be generalized to \(b_1 \ne 1\), but they would differ from the classical theory, object of this paper.

  2. For the relation between the Möller and the Buchberger–Möller algorithms see Sect. 9.

  3. Möller’s algorithm works in the same way for any term-ordering; Cerlienco-Mureddu applies only for the lexicographical term-ordering.

  4. For infinite sets of monomials, it is possible to have an associated Bar Code, provided that such sets are order ideals.

  5. They are related to the primary defining set of the investigated code.

  6. A corner cut is the monomial linear basis which is the support of all normal forms modulo an ideal given by a Gröbner basis and computed via Buchberger reduction.

  7. The non-commutative version of the FGLM Algorithm presented in [12] at our knowledge is the single available non-commutative algorithm in the families stem from [69] unless we decide to enclose also Todd–Coxeter algorithm [96].

  8. For the extension of their theory removing the radicality hypothesis see [72].

  9. Not necessarily consisting of normal forms w.r.t. the Gröbner basis of I w.r.t. a term ordering and not even consisting of monomials.

  10. \(\delta :=\#{{\mathfrak a}} = \#{{\mathfrak b}}.\)

  11. \(\overline{T}\) cannot contain repeated terms, while the \(\overline{T}^{[i]}\), for \(1<i \le n\), can. In case some repeated terms occur in \(\overline{T}^{[i]}\), \(1<i \le n\), they clearly have to be adjacent in the list, due to the lexicographical ordering.

  12. Clearly if a term \(\pi ^i(t_{\overline{j}})\) is not repeated in \(\mathfrak {M}^{[i]}\), the sublist containing it will be only \([\pi _i(t_{\overline{j}})]\), i.e. \(h=0\).

  13. The label depends on the position in the Bar Code, so if we insert a new bar \(\mathsf{{B}}_{j+1}^{(h)}\), the bars on its right change label becoming \(\mathsf{{B}}_{j+2}^{(h)}\) and so on.

  14. These are the points \(P_i\), \(1 \le i < N\) such that \(\pi ^{h}(\alpha ^{(N)})=\pi ^{h}(\alpha ^{(i)}) = (\alpha _h^{(N)},0,\ldots ,0)\), similarly to Cerlienco–Mureddu algorithm. The only difference is that we do not consider \(P_N\).

  15. Not necessarily a monomial one.

  16. Each \(r_i\) is the product of \(N-1\) factors \(p_{ik}^{[c_{i,k}]}=\frac{1}{a_{c_{i,k},i}-a_{c_{i,k},k}}\cdot (x_{c_{i,k}}- a_{c_{i,k},k})\).

  17. The adjective “primitive” only means that the code is over \(\mathbb {F}_{q^m}\) and \(n=q^m-1\).

  18. Indeed the last point has exponent \(2i+1=2(2^{m-1}-1)+1=2^{m}-1=n\). Note also that \(\vert \{0,\ldots ,2^{m-1}-1\}\cup \{-\infty \} \vert =2^{m-1}+1\).

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Authors and Affiliations

  1. Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy

    Michela Ceria

  2. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Samuel Lundqvist

  3. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genoa, Italy

    Teo Mora

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Notational choices

Notational choices

We summarize here our notational choices within the paper.

a, c Field elements

\(\alpha\) Root

A, M Square matrices

\(\mathbf{a},\mathbf{b},\mathbf{c}\) Basis of an algebra

\({{\mathfrak a}},{{\mathfrak b}}\) Monomial basis of an algebra

\(\mathcal {A}\) Algebra

b Algebra elements

\(\mathsf {B}\) Bar code

\(D,\delta ,\gamma ,k,m,n, \nu , N,s,t\) Integers

d Dixon polynomial

\(\Delta\) Dixon resultant

\(\mathsf{D}\) Dixon matrix

\(\mathcal {E}\) Vector space

\({{\mathcal {F}}}\) Set of polynomials

\(f,g,\overline{\gamma },h,U,V,v\) Polynomials

\(\mathbf{f}\) Module element

\({{\bar{f}}}\) Distributed cost

F Dynamical system representation

\(\overline{F}\) Matrix whose rows are vectors

\(\Psi\) Endomorphism

\(\phi\) Functional

\(\Phi\) Cerlienco–Mureddu correspondence

\(\mathbf{g}\) Number of elements of a Gröbner basis

\(\mathcal {G}\) a Gröbner basis

\(\mathbf {k}\) Field

I Ideal

J Semigroup ideal

E, L List

\(\mathfrak {M}\) Matrix of terms

\(\mathsf {N}\) Normal set

\(\mathcal {N}\) Module

P Points

\(\mathcal {P}\) Ring

\(\mathcal {Q}\) Family of separators

Q, R Separators

S Set of points

\({\mathcal {S}}\) Linearly indipendent set of polynomials

\(t,\tau\) Monomial

T Set of monomials

T Leading term

\(\mathcal {T}\) Semigroup of terms

\(\mathfrak {T}\) Trie

\(u,w\) Vertices of a tree

\(\bar{e}\),\(\bar{u}\), \(\bar{v}\),\(\bar{w}\) Vectors

\(\mathcal {V}\) Vector space

x, y, z, T Variables

\(\chi\) Characteristic polynomial

\(\mathbf{X}\) Set of points

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Ceria, M., Lundqvist, S. & Mora, T. Degröbnerization: a political manifesto. AAECC 33, 675–723 (2022). https://doi.org/10.1007/s00200-022-00586-z

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