Singularities of Amoeba Contours

We give explicit real-algebraic equations for computing the singularities of amoeba contours, separating degeneracies of the logarithmic critical locus from coincidences of distinct critical lifts. For plane curves, this yields practical systems detecting nodes, cusps, and multiple branches. We also show that maximal sparsity does not maximize contour singularities, even for lattice triangles and parallelograms.