All Isolated Complex Roots of a Polynomial System via Homotopy Continuation¶
- Task ID:
math.homotopy_poly_roots - Domain:
math - Subdomain:
computational_science - Status:
test - Tags:
numerical_methods,algebraic_geometry,homotopy_continuation,polynomial_systems,predictor_corrector,path_tracking,singular_roots,divergent_paths
Public Summary¶
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Example B1 Prompt Excerpt¶
You are a numerical methods expert. Your goal is to compute **all isolated complex roots** of a bivariate polynomial system \(F(x)=0\) with \(x=(x_0,x_1)\in\mathbb{C}^2\) using **homotopy continuation**, and produce diagnostic outputs that prove your solver actually tracks paths.
## Input (read all of these — they define a **single** continuation contract)
- `data/system.json`: target system \(F\) (bivariate), same structure as before:
- `d1`, `d2`: degrees of the two polynomials
- `n_paths`: the Bézout number \(d_1 \cdot d_2\) (total number of paths to track)
- `tolerances`: `match_tol`, `residual_pass_tol`, `real_tol`, `singular_sigma_min_tol`
- `polynomials`: a list of two polynomial objects (`f1`, `f2`), each with `terms`
- Each term has: `coeff_re` (float), `coeff_im` (float), `exponents` (two integers)
Notes¶
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