Characterizations of border bases

Abstract

This paper presents characterizations of border bases of zero-dimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebases is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute.

The Approach of Moments for Polynomial Equations

“…In particular, algorithms have been proposed for constructing border bases of I leading to general (connected to 1) bases of R[x]/I (see [9,Chap. 4], [14], [29], [43]); these objects are introduced below. The moment matrix approach for computing real roots presented in this chapter leads naturally to the computation of such general bases.…”

“…This is indeed the case, for instance, for the prolongation-projection algorithm of [35] which, as shown in [19], can be turned into an algorithm for real roots by adding a positivity condition. The algorithm of [35] works with the space K C t but uses a different stopping criterion instead of the rank condition (14). Namely one should check whether, for some D ≤ s ≤ t, the three affine spaces π s (K C t ), π s−1 (K C t ), and π s (K C t+1 ) have the same dimensions (where π s (Λ) denotes the restriction of Λ ∈ (R[x] t ) * to (R[x] s ) * ); if so, one can compute a basis of R[x]/I and extract V C (I).…”

Border bases and kernels of homomorphisms and of derivations

“…This ideal is a monoid ideal, not a ring ideal. The border of is the set of terms ∂ = ( There are many equivalent conditions to the above definition; eighteen of them are given in [3]. We will use the condition of Proposition 3.1, called the Buchberger criterion for border bases.…”

A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

“…Other techniques have been proposed for producing bases of the ideal I and of the vector space R[x]/I, which do not depend on a specific monomial ordering. In particular, algorithms have been proposed for constructing border bases of I leading to general (stable by division) bases of R[x]/I (see [6,Chapter 4], [8] and [23]). Another normal form algorithm is proposed by Mourrain [14] (see also [15,17]) leading to more general (namely, connected to 1) bases of R[x]/I.…”