• Corpus ID: 59316430

Signature-based Möller's algorithm for strong Gröbner bases over PIDs

@article{Francis2019SignaturebasedMA,
  title={Signature-based M{\"o}ller's algorithm for strong Gr{\"o}bner bases over PIDs},
  author={Maria Francis and Thibaut Verron},
  journal={ArXiv},
  year={2019},
  volume={abs/1901.09586}
}
Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors. In this paper, we present a signature-based version of M\"oller's classical variant of Buchberger's algorithm for computing strong Gr\"obner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease… 
View PDF on arXiv
1 Citation
Background Citations
1

Tables from this paper

One Citation

Efficient Gröbner bases computation over principal ideal rings

21 References

Signature-based Criteria for Möller's Algorithm for Computing Gröbner Bases over Principal Ideal Domains

A signature-based algorithm for computing Gr\"obner bases over principal ideal domains (e.g. the ring of integers or theRing of univariate polynomials over a field) is presented, adapted from M\"oller's algorithm (1988) which considers reductions by multiple polynmials at each step.

A survey on signature-based algorithms for computing Gröbner bases

On Signature-Based Gröbner Bases Over Euclidean Rings

A hybrid algorithm is presented trying to combine the advantages of signature-based and non-signature-based Gröbner basis computation, and speedups are achieved due to faster finding good reducers with the hybrid technique.

Signature-based algorithms to compute Gröbner bases

A Buchberger-style algorithm to compute a Gröbner basis of a polynomial ideal, allowing for a selection strategy based on "signatures" is described, leading to some surprising results.

A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)

A new efficient algorithm for computing Gröbner bases is introduced that replaces the Buchberger criteria by an optimal criteria and it is proved that the resulting algorithm (called F5) generates no useless critical pairs if the input is a regular sequence.

On an Installation of Buchberger's Algorithm

Computing a Gröbner Basis of a Polynomial Ideal over a Euclidean Domain

On ideal lattices, Grobner bases and generalized hash functions

This paper defines a worst case problem, shortest substitution problem with respect to an ideal in ℤ[x1,…,xn], and uses its computational hardness to establish the collision resistance of the hash functions.

Practical Gröbner basis computation

This work investigates signature based algorithms in detail and introduces new practical data structures and computational techniques for use in both signature based Gröbner basis algorithms and more traditional variations of the classic Buchberger algorithm.

A criterion for detecting unnecessary reductions in the construction of Groebner bases

A new criterion is presented that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals and allows to derive a realistic upper bound for the degrees of the polynomials in the GroBner-Bases computed by the algorithm in the case of poylemials in two variables.