The Galois/monodromy group of a parametrized polynomial system describes how the solutions permute as parameters traverse loops in parameter space. A system with n solutions has monodromy group contained in the symmetric group Sₙ, and the actual group — which specific permutations are realized — encodes the algebraic structure of the system.

Computing this group numerically is standard: track solution paths along loops using homotopy continuation, record which solutions permute into which, and accumulate permutations until the group is generated. The problem is certification. Numerical path tracking is approximate — floating-point arithmetic accumulates error, and if two solution paths come close together, the tracker might confuse them. A wrong permutation corrupts the entire group computation.

Certified homotopy path tracking guarantees correctness by maintaining rigorous error bounds throughout the path. At each step, the algorithm produces not just an approximate solution but an interval containing the true solution. If two intervals ever overlap, the step size is refined until they separate. The permutation at the end of a loop is certified because each solution has been tracked with guaranteed identity preservation.

The certification turns a heuristic computation into a proof. The output is not “the monodromy group is probably G” but “the monodromy group contains these permutations, verified by certified computation.” Generating the full group still requires checking enough loops, but each individual permutation is exact.

The experiments validate known results and discover new ones. For systems arising in kinematics, algebraic statistics, and enumerative geometry, the certified monodromy groups match theoretical predictions where available and provide new data where predictions don’t exist.

The path doesn’t just approximate the answer — it proves the answer. The certificate is the computation itself, done with sufficient care.