Mechanism guaranteeing return of zeros to the real axis after collisions in the A-variation framework

Establish a general mechanism that guarantees the return of zeros to the real axis, after they move off the real line due to collisions, when varying the generalized sections ZN(t; ā) along paths in the A-parameter space 𝒁N connecting the core Z0(t) to ZN(t; 1̄).

Background

In demonstrating the A-variation method, the paper shows that along a simple linear path in parameter space, pairs of real zeros can collide and become complex conjugate, and later (empirically) return to the real axis.

While a specific non-colliding path can be engineered, the authors note the absence of a general theoretical mechanism ensuring that zeros which become non-real will subsequently return to the real axis, highlighting a key gap in the variational approach.