Non-constant coefficient tropical hypersurfaces

#Draw the Newton polytope and tropical hypersurface of the tropical polynomial $$f = 1 \oplus x \oplus (1 \odot x^2) \oplus y \oplus (x \odot y) \oplus 1 \odot y^2 .$$

  1. Consider the "lifted" Newton polytope of $f$: the convex hull of the points $(a,b,c)$ where $a \odot x^b \odot y^c$ is a monomial in $f$. Projection of its lower faces (those whose inward-point normal vectors have positive first coordinate) gives a regular subdivision of the Newton polytope of $f$. Draw this subdivision.
  2. What is the relationship between the subdivision and the tropical hypersurface?


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