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

Let Z denote a finite collection of reduced 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 whether the points in Z are in (i,j)-uniform position. In addition it can be used to determine whether the points in Z are in uniform position, in linearly general position and in general position. The algorithm uses the Chow form of various d-upleembeddings of Z and derivatives of these forms. The existence of the algorithm provides an answer to a question of Kreuzer.