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All Isolated Complex Roots of a Polynomial System via Homotopy Continuation

  • Task ID: math.homotopy_poly_roots
  • Domain: math
  • Subdomain: computational_science
  • Status: final
  • Benchmark set: seed42 (60 tasks)
  • Tags: numerical_methods, algebraic_geometry, homotopy_continuation, polynomial_systems, predictor_corrector, path_tracking, singular_roots, divergent_paths

Runtime and requirements

  • Estimated time: 60-120 minutes
  • Python: >=3.11
  • Packages: numpy>=1.26, scipy>=1.11, sympy>=1.12
  • GPU required: no
  • Network required: no

Public input and output contract

Inputs

  • system.json (data): Polynomial system coefficients (monomial list form) and numeric tolerances
  • start_system.json (data): Instance-fixed total-degree start polynomials g1,g2 and gamma (re,im) for the homotopy
  • start_roots.npy (data): Start points in path row order; path_log.npy must match these rows

Outputs

  • solver.py (code): Agent-written homotopy continuation solver
  • roots.npy (data): All path endpoints (one per tracked path; converged paths contain root, diverged paths contain sentinel 1e8+1e8j)
  • residuals.npy (data): Final residual norm ||F(x)||_2 for each path endpoint
  • is_singular.npy (data): Singularity flag per root (True if Jacobian is rank-deficient at that root)
  • is_real.npy (data): Real-root flag per root (True if max|Im(x_i)| <= real_tol)
  • condition_numbers.npy (data): Condition number of Jacobian at each root (sigma_max/sigma_min)
  • path_log.npy (data): Start points of homotopy paths (one per tracked path)
  • path_status.npy (data): Convergence status per path: 0=converged to finite root, 1=diverged to infinity

Public repository files

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Provenance

This task is included in the current seed42 benchmark set at dataset revision f11a199f71fb4b854ac43a1bf548d5df519141a9.

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