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We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger-Moeller algorithm, best suited for the computation over QQ, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in cocoa, and we report some experimental timings

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.

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.

Abstract. A set of s points in f4 is called a Cayley-Bacharach scheme (CBscheme), if every subset of s -1 points has the same Hubert function. We investigate the consequences of this "weak uniformity." The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hubert function of a CB-scheme X has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a CB-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hubert functions of Cohen-Macaulay domains.

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