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Robust Numerical Algebraic Geometry

Emma R. Cobian, Jonathan D. Hauenstein, Charles W. Wampler

TL;DR

This work develops a fiber-product–based robustness framework for numerical algebraic geometry to address intrinsic uncertainty in parameterized polynomial systems. By enforcing algebraic conditions on multiple solution points simultaneously, the method locates nearby parameters p* where the solution set exhibits well-conditioned, intended structures (e.g., fewer finite solutions, emergent/infinite behavior, irreducible decompositions, or higher multiplicities). The approach is demonstrated on a suite of kinematic and robotic problems (e.g., decomposable 4-bar, Stewart-Gough, and 6R inverse kinematics), using gradient-descent homotopies with homogenized Lagrange multipliers and, when needed, isotropic coordinates. Key tools include witness sets, Macaulay matrices, local Hilbert functions, and trace tests, all integrated via fiber products to yield a practical and scalable framework for robust interpretation of numerical results under uncertainty. The results underscore the potential of nearby-exception analysis to stabilize computations and reveal meaningful parameter regimes in applied geometric computation.

Abstract

The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty may arise in the structure of the solution set, including the number of isolated solutions, the existence of higher dimensional solution components, and the number of irreducible components along with their multiplicities. The loci where these structures change form a stratification of exceptional algebraic sets in the space of parameters. We describe methodologies for making the interpretation of numerical results more robust by searching for nearby parameter values on an exceptional set. We demonstrate these techniques on several illustrative examples and then treat several more substantial problems arising from the kinematics of mechanisms and robots.

Robust Numerical Algebraic Geometry

TL;DR

This work develops a fiber-product–based robustness framework for numerical algebraic geometry to address intrinsic uncertainty in parameterized polynomial systems. By enforcing algebraic conditions on multiple solution points simultaneously, the method locates nearby parameters p* where the solution set exhibits well-conditioned, intended structures (e.g., fewer finite solutions, emergent/infinite behavior, irreducible decompositions, or higher multiplicities). The approach is demonstrated on a suite of kinematic and robotic problems (e.g., decomposable 4-bar, Stewart-Gough, and 6R inverse kinematics), using gradient-descent homotopies with homogenized Lagrange multipliers and, when needed, isotropic coordinates. Key tools include witness sets, Macaulay matrices, local Hilbert functions, and trace tests, all integrated via fiber products to yield a practical and scalable framework for robust interpretation of numerical results under uncertainty. The results underscore the potential of nearby-exception analysis to stabilize computations and reveal meaningful parameter regimes in applied geometric computation.

Abstract

The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty may arise in the structure of the solution set, including the number of isolated solutions, the existence of higher dimensional solution components, and the number of irreducible components along with their multiplicities. The loci where these structures change form a stratification of exceptional algebraic sets in the space of parameters. We describe methodologies for making the interpretation of numerical results more robust by searching for nearby parameter values on an exceptional set. We demonstrate these techniques on several illustrative examples and then treat several more substantial problems arising from the kinematics of mechanisms and robots.
Paper Structure (28 sections, 6 theorems, 51 equations, 10 figures, 14 tables)

This paper contains 28 sections, 6 theorems, 51 equations, 10 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Robustness scenarios
  3. Multiplicity example
  4. Divergent solutions
  5. Emergent solutions
  6. Sets of exceptional dimension
  7. Exceptional decomposition
  8. Exceptional multiplicity
  9. Summary
  10. Robustness framework
  11. Projective space and solutions at infinity
  12. Solutions at infinity
  13. Illustrative example
  14. Witness points and randomization
  15. Witness points
  16. ...and 13 more sections

Key Result

Theorem 3.1

Suppose that $f(x;p)$ is a polynomial system and $A\subset V(f(x,p))\subset{\mathbb C}^n\times{\mathbb C}^m$ is an irreducible algebraic set. For auxiliary variables $y$ and constants $c$ used for randomization and slicing, let $F_c(x,y,p)$ be a polynomial system which imposes a condition on the par Let $\Delta_a\subset V({\mathcal{F}}_a)$ be an algebraic set of components to ignore such that ther

Figures (10)

  • Figure 1: Solution sets for the initial and perturbed parameters
  • Figure 2: (a) Illustration of recovering parameters for various perturbations including the example summarized in Table \ref{['table:PosDim_Parmaeters']}; (b) Illustration using 500 samples
  • Figure 3: Histograms for (a) $p_1$, (b) $p_2$, and (c) intrinsic parameterizing coordinate for recovered parameter values from $500$ samples
  • Figure 4: Solution sets corresponding to the (a) initial and recovered parameters, and (b) perturbed parameters
  • Figure 5: (a) Illustration of recovering parameters for various perturbations including the example summarized in Table \ref{['table:Multiplicity_Parmaeters']}; (b) Illustration using 500 samples
  • ...and 5 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • Remark 3.4
  • Corollary 4.1
  • Corollary 5.1
  • Corollary 6.1
  • Corollary 7.1
  • proof