James J. Madden: Algorithms for solving infinitely near base conditions

Abstract: Let P₀ be a point in affine n-space, and let { P_i } be a sequence of points with P_{i + 1} in the exceptional divisor of the blow-up of P_i . We seek to compute the dimension of the linear system of hypersurfaces of some bounded degree that pass through the points P_i with some specified multiplicities. This is essentially just a problem in linear algebra, but there is a large amount of data that one needs to keep track of. When n = 2, the problem is solved very efficiently by Zariski's theory of complete ideals. We are looking to generalize this theory sufficiently to make tools useful in the context of the Pierce-Birkhoff problem in real algebraic geometry.