A generalization of the FGLM technique is given to compute Grr obner bases for two-sided ideals of free nitely generated algebras. Specializations of this algorithm are presented for the cases in which the ideal is determined by either functionals or monoid (group) presentations. Generalizations are discussed in order to compute G-bases on (twisted) semigroup rings. It is well known that the co...

It is well known that the complexity of Gröbner bases computation strongly depends on the term ordering, moreover, elimination orderings often yield a greater complexity. This remark led to the so-called FGLM conversion problem, i.e. given a Gröbner basis w.r.t. a certain term ordering,‖ find a Gröbner basis of the same ideal w.r.t. another term ordering. One of the efficient approaches for sol...

Nowadays, many strategies to solve polynomial systems use the computation of a Gröbner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gröbner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show ...

We consider matrices over a ring K [∂; σ, θ ] of Ore polynomials over a skew field K . Since the Popov and Hermite normal forms are both Gröbner bases (for term over position and position over term ordering resp.), the classical FGLM-algorithm provides a method of converting one into the other. In this report we investigate the exact formulation of the FGLM algorithm for not necessarily “zero-d...

We show herein that a pattern based on FGLM techniques can be used for computing Gröbner bases, or related structures, associated to linear codes. This Gröbner bases setting turns out to be strongly related to the combinatorics of the codes.

PURPOSE A prior study showed that a tetrapeptide (FGLM-amide) derived from the carboxyl terminus of substance P (SP) and a 12-residue peptide corresponding to the C domain of insulin-like growth factor (IGF)-1 mimic the synergistic effect of the full-length molecules on corneal epithelial wound healing. To develop an effective treatment for persistent corneal epithelial defects, the current stu...

We compare thoroughly the Berlekamp – Massey – Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multidimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.

Given a zero-dimensional ideal I ⊂ K[x1, . . . ,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several effi...