Isosingular Sets and Deflation

Abstract

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which are similar to the sequences arising in Thom-Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic sets that can be investigated with methods from numerical algebraic geometry.

Numerically intersecting algebraic varieties via witness sets

“…Isosingular theory [10] uses a strong deflation operation that is related to the weak deflation method of [14]. Let JF (x) be the M ×N Jacobian matrix of partial derivatives of F : C N → C M , and suppose x * ∈ V(F ) ⊂ C N .…”

“…For the purposes of this paper, the determinantal form of strong deflation is preferred over other forms given in [10] that introduce auxiliary null-space variables λ. In particular, when auxiliary variables are introduced, the original point x * is a projection of the final point (x * , λ * ), which complicates matters when x * is a point on some positive dimensional set that we wish to subject to further operations.…”

“…It is important to note that there exists a computable test to determine when the stabilization value k * of Item 2 is reached [10]. Item 3 will have important consequences in our treatment of the pair-wise intersection problem.…”

“…Starting with witness sets for A and B, the first step is to track 49 paths using the homotopy H X,1 defined by (10). This yields the sets S 1 consisting of 4 points and N 1 consisting of 40 points.…”

“…To do so, we provide an alternative approach to filtering out junk points and show how to generate the witness systems F ij to complete the irreducible witness sets. The necessary tools are provided by the recently developed theory of isosingular sets [10]. The existing diagonal homotopy suffices as is when all the irreducible components of A ∩ B are also irreducible components of V(F A , F B ).…”