On deflation and multiplicity structure

TL;DR

This paper introduces two novel methods for analyzing singular solutions of polynomial systems: a linear differential form-based deflation technique and a system for determining multiplicity structure, both enabling efficient and exact treatment of roots.

Contribution

It presents a deflation method that avoids variable increase and a new approach to determine multiplicity structure with fewer variables, improving efficiency and exactness.

Findings

The deflation method does not add new variables and scales linearly with iterations.

The new system captures multiplicity structure with a small number of additional variables.

Both methods are exact and suitable for certification of singular roots.

Abstract

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are "exact" in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system.