相关论文: An F5 Algorithm for Tropical Gr\"obner Bases in th…

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear…

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of $K$.Because of the use of the valuation, the theory of tropical Gr{\"o}bner…

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr\"obner bases taking into account the valuation of $K$. Because of the use of the valuation, this theory is promising for stable…

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases…

The F5 algorithm is generally believed as one of the fastest algorithms for computing Gr\"{o}bner bases. However, its termination problem is still unclear. Recently, an algorithm GVW and its variant GVWHS have been proposed, and their…

The signatures of polynomials were originally introduced by Faug\`{e}re for the efficient computation of Gr\"obner bases [Fau02], and redefined by Arri-Perry [AP11] as the standard monomials modulo the module of syzygies. Since it is…

We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$…

A generalized criterion for signature related algorithms to compute Gr\"obner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gr\"obner basis, including the famous F5 algorithm,…

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known…

Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective…

In this paper we give an insight into the behaviour of signature-based Gr\"obner basis algorithms, like F5, G2V or SB, for inhomogeneous input. On the one hand, it seems that the restriction to sig-safe reductions puts a penalty on the…

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights…

A generalized criterion for signature-based algorithms to compute Gr\"obner bases is proposed in this paper. This criterion is named by "generalized criterion", because it can be specialized to almost all existing criteria for…

The famous F5 algorithm for computing Gr\"obner basis was presented by Faug\`ere in 2002 without complete proofs for its correctness. The current authors have simplified the original F5 algorithm into an F5 algorithm in Buchberger's style…

We present a generic and executable formalization of signature-based algorithms (such as Faug\`ere's $F_5$) for computing Gr\"obner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are…

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free…

In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the…

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gr\"obner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for…

We consider the problem of computing a grevlex Gr\"obner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular…

Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions…