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

Abstract

Isolated singularities typically occur at self-intersection points of planar algebraic curves, curve offsets, intersections between spatial curves and surfaces, and so on. The information characterizing the singularity can be captured in a local dual basis, expressing combinations of vanishing derivatives at the singular point. Macaulay's algorithm is a classic algorithm for computing such a basis, for a point in an algebraic set. The integration method of Mourrain constructs much smaller matrices than Macaulay's approach, by performing integration on previously computed elements. 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 behaviour 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.

Bivariate polynomial reduction and elimination ideal over finite fields

“…4.1) in a more general multivariate context. For a study of differences between the resultant and the generator of the elimination ideal the reader may refer to (Mantzaflaris et al, 2016).…”

Analyzing the dual space of the saturated ideal of a regular set and the local multiplicities of its zeros

“…Hauenstein and others 3 proposed an algorithm to compute the coefficients of the so‐called inverse system or dual basis, which defines the multiplicity structure at the singular root. Mantzaflaris and others 4 introduced the notion of directional multiplicity, which expresses the multiplicity structure with respect to an axis and is useful in understanding the geometry behind projection. Furthermore, the computation of multiplicity structure could be reduced to solving eigenvalues of the so‐called multiplicity matrix, which is studied by Möller and Stetter 5,6 and others 7,8 …”

Subresultants and the Shape Lemma

“…It is clear that both multiplicities and roots at infinity play a decisive role here. In [MRZ16], directional multiplicities are used to explain the differences between the degrees of these two polynomials. In [GRZ13], the connection between the elimination ideal and univariate resultants of two generators is explored.…”