A symbolic test for (i,j)-Uniformity in reduced zero-schemes

Let Z denote a finite collection of points in projective n-space and let I denote the homogeneous ideal of Z. The points in Z are said to be in (i, j)-uniform position if every cardinality i subset of Z imposes the same number of conditions on forms of degree j. The points are in uniform position if they are in (i, j)-uniform position for all values of i and j. We present a symbolic algorithm that, given I, can be used to determine if the points in Z are in (i, j)-uniform position. In addition it can be used to determine if the points in Z are in uniform position, in linearly general position and in general position. The algorithm uses the Chow Form of various dUple embeddings of Z and derivatives of these forms. The existence of the algorithm provides an answer to a question of Kreuzer.