The first Gröbner basis algorithm was constructed by Buchberger in 1965; thus it bears his name to this day – Buchberger’s algorithm.[9] Though Buchberger’s algorithm looks relatively simple, it can take a very large amount of time. The step that creates 0 h via a normal form calculation is computationally very difficult. This is particularly frustrating (and wasteful) if the normal form calcul...

When Zharkov and Blinkov ([ZB93]) applied the classical ideas of involutive systems originating from the theory of partial dierential equations (c.f. [Ja29],[Po78]) to the computation of Gr obner bases (c.f. [Bu65],[BW93]) their theory seemed to be a rather marginal concept. But due to the opportunity of gaining a faster version for one of the most frequently applied algorithms the method came...

For a class of cubic differential systems, Kukles established necessary and sufficient conditions for the existence of a centre. This paper presents a particular system that shows that either Kukles' conditions are incomplete, or the multiplicity of the origin as a focus is greater than or equal to 9. The second author described in [6] a mechanical procedure based on Poincare's classic method, ...

In this paper, two protocols are presented for private intersection detection of moving objects’ trajectories. Assuming that the movement trajectory an object can be described by a time-dependent polynomial function, problem finding points is simplified to common roots corresponding polynomials. Thereafter, GrObner Basis used design novel secure protocol Another also designed based on distance ...

Nous étudions la minimisation de la consommation d’énergie des réseaux sans-fil fixes à transmission par liens microondes, par l’optimisation jointe du routage des flux de données et la sélection de la configuration des liens. Nous présentons une formulation mathématique exacte basée sur un multiflot entier de coût minimum avec des fonctions de coût en escalier, rendant le problème très diffici...

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal I in a polynomial ring a monomial ideal in≺I, in some special situations the monomial ideal in≺I is square free. On the other hand given any monomial ideal I of a polynomial ring S, we can define the toric K[I] ⊂ S. In this paper we will study toric rings defined by Segre embed...

Quantum sensors may provide extremely high sensitivity and precision to extract key information in a quantum or classical physical system. A fundamental question is whether sensor capable of uniquely inferring unknown parameters system for given structure the admissible measurement on sensor. In this paper, we investigate capability class which consist either single qubit two qubits. coupled sp...