The vanishing ideal of a finite set of closed points in affine space

Abstract

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.

The Vanishing Ideal of a Finite Set of Points with Multiplicity Structures

“…That's why we began to study the vanishing ideal of the set of points with multiplicity structures which is essentially a special case of Birkhoff interpolation problem. I still remember the moment when I first read the paper [2] written by Mathias Lederer in which the quotient basis and Gröbner basis can be got in a geometric way. I told myself that this was just what we wanted.…”

“…Although whether or not the points are with multiplicity structures matters much, the paper really inspired us a lot. Our algorithm also uses induction over variables and the definition of addition of lower sets is essentially the same with that in paper [2]. However during the induction procedure, we have to consider p-Special case and p-General case.…”

“…This consideration, on one hand, clearly indicates the geometric meaning of the multiplicity structures of points, on the other hand, means a lot for our capacity of applying the algorithm of intersection of two ideals. In paper [2], the author uses Lagrange interpolation method to get the vanishing polynomial of all points from the polynomials vanishing on subsets of the points. However the Lagrange interpolation method could just not work for our problem because the points are with multiplicity structures.…”

Finite sets of d-planes in affine space

“…I wish to thank the anonymous referee of [Led08] for his positive evaluation of my article, which motivated me to carry on my research on this subject. Many thanks go to B. Heinrich Matzat and his group in Heidelberg, in particular Michael Wibmer, who gave me the opportunity to present my research in a seminar talk.…”

“…For a discussion of this issue, see [Led08] and references therein. In the present paper, we want to generalise Eq.…”