Computing Ideals of Points

The easiest geometric object in affine or projective space is a single rational point. It has no secrets, in particular its defining ideal, i.e. the set of all the polynomials which vanish at the point, is straightforward to describe. Namely, for an affine point with coordinates (a1, . . . , an), the corresponding ideal is p = (x1 − a1, . . . , xn − an); while for a projective point with coordinates (a0 : . . . : an), the corresponding homogeneous ideal is B = (aix1 − a1xi, . . . , aixn − anxi) for any i such that ai 6= 0. But obviously the matter becomes more complicated if we want the algebraic description of the vanishing ideal of a finite set of points. Let K be a field, and let AK and PK denote, respectively, the n-dimensional affine and projective spaces over K. Then let X := {P1, . . . , Ps} be a finite set of s rational points; that is, points having coordinates in the field K. Why are we interested in the defining ideal I(X)? And how do we compute it? We answer the first question by listing just a few examples we know of where ideals of points have been used in different fields of mathematics.