Computing border bases

Abstract

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering and hence compute a border basis that contains the reduced -Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.

Border bases and kernels of homomorphisms and of derivations

“…In [4] there are several algorithms for computing border bases presented. We will employ the following Basis Transformation Algorithm ( [4], Proposition 5).…”

“…In some cases border bases algorithms are significantly faster than those of Gröbner bases (see 19 and 20 of [4]). The reason lies in the fact that the border basis computation requires polynomials of much lower degrees (although there are often more polynomials).…”

“…When we compute all the border bases of an ideal, then as a consequence we also easily obtain all the reduced Gröbner bases ( [4]), whereas the reverse is not true.…”

Toric border basis

“…These structures are described effectively by a set of polynomials which represent the normal forms in the quotient structure and a method to compute the normal form of any polynomial. This family of methods includes, for instance, Gröbner basis [3,6] or border basis computation [15,18,12]. A "fixed-point" strategy is involved in these algorithms: starting with the initial set of equations, so-called S-polynomials or commutation polynomials are computed and reduced.…”

“…This monomial ordering is used to define the initial ideal associated to the ideal of the equations. The border basis approach extends Gröbner basis methods by removing the monomial ordering constraint, which may induce numerical instability when the coefficients of the polynomials are known approximately [15,18,11,13,12,16,19,10,20].…”