Certifying isolated singular points and their multiplicity structure

TL;DR

This paper introduces a new deflation method for isolated singular roots of polynomial systems that is more efficient and provides a way to determine the multiplicity structure using a small set of equations.

Contribution

It presents a novel deflation technique using a single differential form and a new approach to compute the multiplicity structure with fewer variables and equations.

Findings

The deflation method does not increase variables and has linear growth in equations.

The new system accurately captures the multiplicity structure at singular roots.

Both constructions are exact and suitable for certification procedures.

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 construc-tion 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 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 vari-ables. We show that the roots of this new system include the original singular root but now with multiplicity one, and the new variables uniquely determine the multiplicity structure. Both constructions are "exact", meaning 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.