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On the solutions of deformed algebraic systems

S. Tanabe, M. N. Vrahatis

TL;DR

This study addresses how the real roots of a system of polynomial equations respond to small deformations. It develops a perturbation bound and a homotopy framework that guarantee the real root count inside a fixed compact set remains unchanged, counting multiplicities, when the perturbation parameter $t$ satisfies $t < \frac{1}{\|\varphi\|\,C(a)\,\mu^2}$. It also establishes a Bertini-Sard-type result ensuring that multiple roots can be decomposed into simple roots via an appropriate perturbation vector $H$, with explicit low-dimensional examples illustrating root-splitting and symmetry-preserving behavior. The results have practical impact for reliable root counting and topology-preserving deformation of real algebraic sets, with potential applications in computer-aided design and graphics, where intersections and perturbations of hypersurfaces are common.

Abstract

A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.

On the solutions of deformed algebraic systems

TL;DR

This study addresses how the real roots of a system of polynomial equations respond to small deformations. It develops a perturbation bound and a homotopy framework that guarantee the real root count inside a fixed compact set remains unchanged, counting multiplicities, when the perturbation parameter satisfies . It also establishes a Bertini-Sard-type result ensuring that multiple roots can be decomposed into simple roots via an appropriate perturbation vector , with explicit low-dimensional examples illustrating root-splitting and symmetry-preserving behavior. The results have practical impact for reliable root counting and topology-preserving deformation of real algebraic sets, with potential applications in computer-aided design and graphics, where intersections and perturbations of hypersurfaces are common.

Abstract

A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.

Paper Structure

This paper contains 5 sections, 6 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The number of roots of a deformed system
  4. Decomposition of multiple roots
  5. Concluding remarks

Key Result

Lemma 3.1

Let us consider $\mu \times \mu$ matrix $A = \left( a_{ij} \right) \in {\rm End}({\mathbb R}^{\mu})$. If $| a_{ij} | < \frac{1}{\mu^2}$, then $({\rm id}_\mu + A)$ is invertible.

Theorems & Definitions (19)

  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Definition 3.1
  • Remark 3.1
  • Lemma 3.1
  • Theorem 3.1
  • Example 3.1
  • Example 3.2
  • Theorem 3.2
  • ...and 9 more