When a Wavefront Becomes Degenerate

This presentation investigates the critical points of wavefronts generated by linear combinations of the four third-order Zernike modes. The main goal is to understand when critical points become degenerate and when they form continuous curves rather than isolated points. Using computational algebra and real algebraic geometry, we eliminate the spatial variables to identify the coefficient combinations that produce degenerate critical points. Its real radical admits a prime decomposition with two projective components: a pure-trefoil line and a twisted cubic. Although both correspond to degenerate wavefronts, they produce different critical-point structures. The pure-trefoil line gives a unique isolated degenerate point, while the twisted cubic characterizes precisely the coefficient configurations producing nonisolated critical sets, each consisting of two parallel Morse-Bott lines. This principal finding is an unexpected connection between Zernike wavefront degeneracy and the classical twisted cubic in real projective three-space.