Computing singular solutions of polynomial systems: towards superlinear convergence without deflation

What the math gives to ML

The paper provides a concrete way to continue an implicitly defined solution through a point where the ordinary Jacobian becomes singular, without augmenting the system with higher derivatives. Its transferable asset is the combination of pseudo-arclength continuation, Jacobian-only correction, and a fractional-power Puiseux predictor that models the non-smooth local geometry of a branch. This suggests a robust continuation wrapper for implicit neural layers or equilibrium models whose fixed-point branch approaches a fold, cusp, or rank-deficient state. The first useful test is whether this predictor permits larger continuation steps and fewer failed Newton solves than linear prediction during homotopy-based training or inference.