Abstract
Farr-Gao algorithm is a state-of-the-art algorithm for reduced Grobner bases of vanishing ideals of finite points, which has been implemented in Maple as a build-in command. This paper presents a two-dimensional improvement for it that employs a preprocessing strategy for computing reduced Grobner bases associated with tower subsets of given point sets. Experimental results show that the preprocessed Farr-Gao algorithm is more efficient than the classical one.
Highlights
Let F be a field, and let Πd := F[x1, x2, . . . , xd] denote the d-variate polynomial ring over F
It is well known that the set of polynomials in Πd that vanish at a finite nonempty set Ξ ⊂ Fd forms an ideal in Πd which is called the vanishing ideal of Ξ, denoted by I(Ξ)
The most significant milestone of computing vanishing ideals is the Buchberger -Moller algorithm [11] that yields, for fixed Ξ and monomial order ≺ on Πd, the reduced Grobner basis G and the Grobnerescalier M of I(Ξ) w.r.t. ≺. It produces a Newton interpolation basis N for the F-linear space spanned by M
Summary
Let F be a field, and let Πd := F[x1, x2, . . . , xd] denote the d-variate polynomial ring over F. ≺. The most significant milestone of computing vanishing ideals is the Buchberger -Moller algorithm [11] that yields, for fixed Ξ and monomial order ≺ on Πd, the reduced Grobner basis G and the Grobnerescalier M of I(Ξ) w.r.t. These results lead to our main algorithm and the timings of some experiments are given
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