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Persistent components in Canny's Generalized Characteristic Polynomial

Gleb Pogudin

Abstract

When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interest. For removing these components, J.M. Rojas proposed to take the greatest common divisor of the results of two different perturbations. In this paper, we investigate this construction, and show that the extra components persistent under taking different perturbations must come either from singularities or from positive-dimensional fibers.

Persistent components in Canny's Generalized Characteristic Polynomial

Abstract

When using resultants for elimination, one standard issue is that the resultant vanishes if the variety contains components of dimension larger than the expected dimension. J. Canny proposed an elegant construction, generalized characteristic polynomial, to address this issue by symbolically perturbing the system before the resultant computation. Such perturbed resultant would typically involve artefact components only loosely related to the geometry of the variety of interest. For removing these components, J.M. Rojas proposed to take the greatest common divisor of the results of two different perturbations. In this paper, we investigate this construction, and show that the extra components persistent under taking different perturbations must come either from singularities or from positive-dimensional fibers.
Paper Structure (6 sections, 8 theorems, 36 equations)

This paper contains 6 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Formal setup and problem statement
  4. Main results
  5. Examples
  6. Proofs

Key Result

Proposition 1

In the notation above, let $C \subset X$ be a proper component. Then $\mathop{\mathrm{GCP}}\nolimits_{\mathbf{x}, \mathbf{p}}(f_1, \ldots, f_n)$ vanishes on $\pi(C)$ for every admissible $\mathbf{p}$.

Theorems & Definitions (27)

  • Definition 1: Resultant
  • Definition 2: Generalized characteristic polynomial
  • Proposition 1
  • Example 1
  • Lemma 1
  • proof
  • Definition 3: Perturbed resultant
  • Remark 1: Computing perturbed resultant
  • Theorem 1
  • Proposition 2
  • ...and 17 more