Gröbner Bases

A Modular Method for Computing the Splitting Field of a Polynomial

“…Reciprocally, the next proposition gives the characterization of all the Gröbner bases of M. Its proof is immediate (see [5]). …”

“…, α n . In this setting, computing K f means to compute a Gröbner basis G of M (see [5]). If we choose the lexicographic order ≺ on terms with x 1 ≺ · · · ≺ x n , then the reduced Gröbner basis of M coincides with the generating set {g 1 , g 2 , .…”

“…For a Gröbner basis G ⊂ Q[X] and a polynomial P , let NF(P, G) denote the normal form of P in Q[X] with respect to G (see [5]). The group S n acts naturally on Q[X] with x σ i = x i σ for 1 i n and σ ∈ S n .…”

On the complexity of the integral closure

“…The set S will arise from division relations among the g i 's and therefore the coefficients of S will be generated in degrees at most ∆ = max{deg g i }. According to [4,Appendix], and especially [11,Theorem B], ∆ is bounded by a polynomial in δ of degree a n (with a in the order of √ 3). This is obviously much larger than the bounds of Theorems 6.3, 6.5 and 6.6.…”

Intersections, ideals, and inversion

“…Given an arbitrary ideal I we may introduce notation symbolizing the derivation of a radical ideal from I , the ideal containing the 'roots' of any polynomials of the form Algorithms are available for extracting the radical of a given ideal (Gianni et al 1989, Eisenbud et al 1992, Becker and Weispfenning 1993 and determining if a polynomial is a member of the radical of a particular ideal (Cox et al 1997, p. 176). Radical ideals are all that is required to produce a one-to-one correspondence between affine varieties (zero sets) and an ideal generated by the defining polynomial equations.…”