Computing Singular Solutions of Polynomial Systems: Towards Superlinear Convergence Without Deflation

Q: Why does homotopy tracking become challenging for singular roots in Numerical Algebraic Geometry?

Tracking the solution curve becomes especially difficult as the parameter approaches zero because the target solution is singular, meaning the Jacobian becomes singular. This creates the “endgame” regime where standard numerical behavior degrades.

Q: What is the key idea of the proposed Arclength Endgame for corank-1 systems?

The method combines classical pseudo-arclength continuation with estimation of the Puiseux series of the curve near the singular root. It achieves a proven superlinear convergence rate using only evaluations of the system and its Jacobian.

Q: How does the work extend estimation of Puiseux series beyond the first exponent ratio?

It introduces a new estimation method that is empirically more stable than prior approaches addressing only the first ratio k1/c. The paper also shows how to estimate ki/c for i ≥ 2.

Jul 16, 2026

Upload by River Wang

Numerical Algebraic Geometry often computes isolated solutions of polynomial systems by tracking a homotopy curve, but this becomes difficult near singular roots in the endgame regime. The work introduces an Arclength Endgame for corank-1 systems, proving superlinear convergence in a neighborhood while using only evaluations of the system and its Jacobian, avoiding additional derivative computations. For higher corank, it proposes a Lifted Arclength Endgame with encouraging experiments. A new, more stable method estimates Puiseux series exponents beyond the first ratio.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages35

Reading Time1 h 28 min

Abstract

Introduction

Problem setup and homotopy tracking

The endgame regime near singular solutions

Corank-1 problems

Main theorem and algorithmic convergence

arXiv :2607 .06329v1 [math .NA] 7 Jul 2026 Computing singular solutions of polynomial systems: towards superlinear convergence without deflation Mikhail Karapetyants Vladimir Kolmogorov Jeferson Zapata Institute of Science and Technology Austria (ISTA) {mikhail. karapetyants,vnk,[jeferson. zapata](jeferson. zapata}@ist. ac. at)[}](jeferson. zapata}@ist. ac. at)[@ist. ac. at](jeferson. zapata}@ist. ac. at) Abstract In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the “endgame” regime) . Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based deflation. We make the following contributions. (1) For corank-1 systems we introduce a new “Arclength Endgame” which combines the idea of the classical pseudo-arclength continuation method with the estimation of the Puiseux series of the curve. We formally prove that it has a superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian, whereas previous techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system. (2) For systems with a larger corank we propose a heuristic “Lifted Arclength Endgame”, which shows promising experimental results. (3) A key step in our approach (as well as in the standard power series endgame) is esti mating the Puiseux series of the curve, which is characterized by fractional exponents ki /c for i ≥ 1 together with associated coefficients. Previous work addressed only estimating the ratio k 1 /c. We present a new method for that which empirically appears to be more stable than previous methods, and also show how to estimate ki /c for i ≥ 2. 1 Introduction We consider the problem of numerically solving a system of equations f (z) = 0 with a zerodimensional set of solutions. Here f is an analytic mapping Cn → Cn. A standard approach in Numerical Algebraic Geometry (NAG) for tackling this problem is as follows. First, one constructsa homotopy function h (z, t) = (1 − t)f (z) + tg(z) where g is a polynomial system with easily computable roots. Consider one such root zroot. If g is chosen generically then there exists a unique smooth function z : (0 , 1] → Cn with z(1) = zroot and h (z(t), t) = 0 for all t ∈ (0 , 1] . Furthermore, if z((0, 1]) is bounded then the limit z ∗ = z(0) = limt→0 z (t) exists and is a root off. The latter condition will always be satisfied if system f is homogeneous. If g is chosen to have sufficiently many roots then every root of f will be covered with probability 1 [SW05, Theorem 8.4.1] . By differentiating equation h (z(t), t) = 0 with respect to t one obtains Davidenko ODE: z˙(t) = −hz (z(t), t)−1ht(z(t), t) (1) We now need to track curve z (t) by numerically solving this ODE. This problem becomes especially challenging when t approaches zero and z ∗ = z(0) is a singular solution z ∗ , i.e. the Jacobian fz(z∗ ) is singular. This regime, referred to as the “endgame”, constitutes the main focus of the present work. The following assumptions are maintained throughout the manuscript: Assumption 1 . (a) h : Cn+1 → Cn is a polynomial mapping, and f (z) = h (z,0) . (b) Point z ∗ ∈ Cn is an isolated solution of f (z) . We denote x∗ = (z∗ , 0) , J ∗ = hz(x∗ ) = fz(z∗ ) and h∗t = ht(x∗ ) . We also let κ = n − rank(J∗ ) be the corank of J ∗ . (c) rank (J∗ ) < n, i. e. z ∗ is a singular solution of f. (d) h∗t is linearly independent of columns in J ∗ , i. e. rank([J∗ |h∗t]) = rank(J∗ ) + 1 . (e) There exists a finite set Π of formal Puiseux series of the form ∞ z (t) = z∗ +X ajtj/c = z ∗ + ak1tk1/c + ak2tk2/c + ... (2) j=1 where c ∈ N and 1 ≤ k1 < k2 < [k](k3... is)[3](k3... is)[...](k3... is)[ is](k3.

FAQs

Why does homotopy tracking become challenging for singular roots in Numerical Algebraic Geometry?

Tracking the solution curve becomes especially difficult as the parameter approaches zero because the target solution is singular, meaning the Jacobian becomes singular. This creates the “endgame” regime where standard numerical behavior degrades.

What is the key idea of the proposed Arclength Endgame for corank-1 systems?

The method combines classical pseudo-arclength continuation with estimation of the Puiseux series of the curve near the singular root. It achieves a proven superlinear convergence rate using only evaluations of the system and its Jacobian.

How does the work extend estimation of Puiseux series beyond the first exponent ratio?

It introduces a new estimation method that is empirically more stable than prior approaches addressing only the first ratio k1/c. The paper also shows how to estimate ki/c for i ≥ 2.

Jul 16, 2026

Upload by River Wang