[Paper Review] Deflation and Certified Isolation of Singular Zeros of Polynomial Systems

Angelos Mantzaflaris, Bernard Mourrain|arXiv (Cornell University)|Jan 17, 2011

Polynomial and algebraic computation12 references4 citations

TL;DR

This paper presents a symbolic-numeric algorithm for certifying singular isolated zeros of polynomial systems by leveraging local ring structure and certified numerical computation. It introduces an improved inverse system computation that reduces intermediate linear system size and derives a one-step deflation technique using differentials, enabling quadratic convergence in Newton solvers and certified uniqueness via interval arithmetic.

ABSTRACT

We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems is presented, which avoids redundant computation and reduces the size of the intermediate linear systems to solve. We derive a one-step deflation technique, from the description of the multiplicity structure in terms of differentials. The deflated system can be used in Newton-based iterative schemes with quadratic convergence. Starting from a polynomial system and a small-enough neighborhood, we obtain a criterion for the existence and uniqueness of a singular root of a given multiplicity structure, applying a well-chosen symbolic perturbation. Standard verification methods, based eg. on interval arithmetic and a fixed point theorem, are employed to certify that there exists a unique perturbed system with a singular root in the domain. Applications to topological degree computation and to the analysis of real branches of an implicit curve illustrate the method.

Motivation & Objective

• To address the challenge of certifying singular isolated roots in polynomial systems, especially when coefficients are approximate or ill-conditioned.
• To develop a method that restores quadratic convergence in Newton-type solvers for multiple roots by deflating the system using multiplicity structure.
• To provide a certified verification of singular roots using symbolic perturbation and interval arithmetic, ensuring existence and uniqueness in a given domain.
• To reduce computational cost in computing inverse systems by avoiding redundant calculations and minimizing intermediate linear system sizes.
• To enable topological analysis of implicit curves and surfaces by computing the number of real branches at singular points.

Proposed method

• The method computes the primal-dual structure of a polynomial system at a singular root using an improved inverse system algorithm that avoids redundant operations.
• It derives a one-step deflation system by applying differential functionals from the dual space to the original equations, based on the multiplicity structure.
• The deflated system is constructed using a certified numerical computation of the dual basis via Mourrain’s integration technique, ensuring accuracy.
• A symbolic perturbation is applied to the system to transform a singular root into an exact multiple root of the perturbed system, enabling certification.
• Verification of the existence and uniqueness of the singular root is performed using interval arithmetic and a fixed-point theorem.
• The method computes the topological degree and number of real branches at a singular point using the trace of the multiplication operator on the dual space.

Experimental results

Research questions

• RQ1 How can we efficiently compute the multiplicity structure of a singular isolated zero of a polynomial system?
• RQ2 What is the most effective way to construct a deflated system that restores quadratic convergence in Newton's method for multiple roots?
• RQ3 How can we certify the existence and uniqueness of a singular root using only numerical approximations and symbolic perturbation?
• RQ4 What is the minimal size of intermediate linear systems required to compute the inverse system, and how can it be reduced?
• RQ5 How can the number of real branches of an implicit curve be computed at a singular point using algebraic invariants?

Key findings

• The proposed inverse system computation reduces the size of intermediate linear systems compared to Macaulay’s method, with a 9×9 matrix in Example 6.2 versus 30×15 in Macaulay’s method.
• In Example 7.1, the method computes a deflated system of 21 equations using a 9×8 matrix, while Macaulay’s method requires a 30×15 matrix.
• For system DZ1 with multiplicity 131, the method uses a 1450×524 matrix, significantly smaller than Macaulay’s 4004×1365 matrix.
• The method successfully computes the number of real branches at a singular point: for Example 6.2, Br(f, (0,0)) = 6, derived from tdeg = 3.
• In Example 7.2, the method computes the multiplicity 18 root with a 206×45 matrix, compared to Macaulay’s 360×165 matrix, showing substantial reduction.
• The method enables partial deflation by stopping at lower-depth structures, yielding a deflated system with only three equations in Example 7.2, verified by a non-zero Jacobian determinant of 64.