Contour Degree of Amoebas of Complete Intersections

We establish the first universal upper bounds for the real degree of the contour of the amoeba of a smooth complete intersection in the algebraic torus.

Our approach extends the Pfaffian method of Lang--Shapiro--Shustin from hypersurfaces to arbitrary codimension by introducing a logarithmic conormal framework based on the logarithmic conormal bundle, the logarithmic Grassmann map, and determinantal rank conditions.

We prove that, on suitable logarithmic conormal charts, the critical locus is locally defined by Schur--complement equations arising from the logarithmic conormal matrix.

This yields explicit universal contour-degree estimates in both regimes $n\ge 2r$ and $n<2r$.

We further replace total-degree arguments by Bernstein's theorem to obtain sparse mixed-volume bounds determined by the Newton polytopes of the transformed equations.

These estimates are frequently much sharper than the corresponding universal bounds and provide a higher-codimensional analogue of the Lang--Shapiro--Shustin theory together with a new geometric interpretation of amoeba contours through logarithmic conormal geometry.