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

Given a finite set of closed rational points of affine space over a field, we give a Gr"obner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger–M"oller 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 Gr"obner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.