Extremal Classification for Pairwise Linear Combinations of Third-Order Zernike Polynomials

We study the extremal behavior of real two-term linear combinations of third-order Zernike modes on the closed unit disk D2. These modes arise naturally in Zernike expansions of optical wavefront aberrations. For each of the six unordered pairwise linear combinations of third-order modes, we classify the interior local extrema in terms of the two real coefficients. The trefoil–trefoil case is treated more generally through linear combinations of primary n-multifoils; harmonicity and the maximum principle show that no interior local extrema occur and that absolute extrema are attained on the boundary circle. For the remaining pairwise combinations, we give analytic conditions for the existence, uniqueness, location, and values of local extrema, including degenerate exceptional cases. We also compare these local extrema with boundary values and describe the associated absolute-extremum problem on the boundary circle. Symbolic computations are included in the appendix to document several algebraic reductions, and numerical illustrations are provided to visualize the resulting classifications.