Cusps and nodes of Amoeba Contours

Abstract

We establish new Newton-polygon bounds for the singularities of amoeba contours of smooth plane curves. Our cusp estimate refines the degree-four bound of Lang--Shapiro--Shustin, reducing its leading coefficient from $8$ to $4$ while retaining the normalized area, boundary lattice points, and directional widths of the Newton polygon. We also obtain a new multiplicity-sensitive bound for transverse $s$-nodes, with the natural decay factor $1/\binom{s}{2}$. The proofs combine logarithmic Gauss maps, normalized fiber products, ramification theory, and saturated off-diagonal intersection schemes, revealing the distinct geometric mechanisms governing cusps and multiple nodes.