Efficient computation of dual space and directional multiplicity of an isolated point

In this work we are interested in the efficiency of dual basis computation, as well as its relation to orthogonal projection. First, we introduce an easy to implement criterion that avoids redundant computations during the computation of the dual basis, by deleting certain columns from the matrices in the integration method. In doing so, we explore general (non-monomial) bases for the associated primal quotient ring. Experiments show the efficient behavior of the improved method. Second, we introduce the notion of directional multiplicity, which expresses the multiplicity structure with respect to an axis, and is useful in understanding the geometry behind projection. We use this notion to shed light on the gap between the degree of the generator of the elimination ideal and the corresponding factor in the resultant.