A numerical method for solving systems of polynomial equations by constructing a homotopy between a start system with known solutions and the target system, then tracking solution paths in the complex domain with predictor–corrector schemes to approximate all isolated roots.

Principle

Exploit continuous deformation (analytic continuation) of solutions: by connecting a system with known solutions to the target via a generic homotopy, isolated solutions of the target correspond to endpoints of tracked paths, subject to path-regularity assumptions.

Demonstration

Construct a total-degree homotopy from a start system whose solutions are all monomially separable; use a predictor step (e.g., Euler) and a corrector (e.g., Newton) to follow each path from t=0 to t=1, producing numerical approximations of all isolated complex roots.

Misapplication

Treating numerical path tracking as a substitute for symbolic multiplicity analysis: naive tracking may fail to resolve singular solutions or may miscount roots when paths diverge or coalesce without deflation or endgame strategies.

Consequence

Provides scalable, often parallelizable numerical approximations of all isolated solutions, enabling numerical algebraic geometry workflows, numerical irreducible decompositions, and verification of solution counts predicted by Bézout-type bounds.

Reversal

Using purely symbolic elimination or resultant-based methods to obtain exact algebraic representations of solutions, foregoing numerical path tracking and its parallel scalability.

Boundary

Targets isolated solutions in the complex domain; performance and reliability depend on choice of homotopy, conditioning of paths, handling of singular or near-singular endpoints, and numerical tolerance settings.

Semantic Tension

Tension exists between numerical homotopy continuation as a robust computational solver and the expectation of exact algebraic information; numerical outputs require certification or symbolic post-processing to claim exact algebraic facts.

Synthesis

Numerical polynomial homotopy continuation follows analytic continuation of solutions from a conveniently solvable start system to the target, using predictor–corrector path tracking and endgame techniques to produce reliable numerical approximations of isolated roots.