Hello, in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR COMPLEX ALGEBRAIC HYPERSURFACES": http://arxiv.org/pdf/math/0205011.pdf There is a lemma about the relation between intersection of a hypersurface with the boundary divisors in a toric variety and the truncation of the defining polynomial of a hypersurface to some face of the newton polytope. (Lemma 2.20. Page 15).

The lemma basically says that If we take a hypersurface with newton polytope P, and consider it's closure in the projective toric variery corresponding the the lattice polytope P, then to find it's intersection with the boundary divisors, it is enough to truncate the polynomial the the corresponding face of the newton polytope and take the zeroes of the truncated polynomial.

The proof given there is a one-liner about the order of vanishing of some monomials and I don't understand why it proves the claim.

I would appreciate if someone could explain this to me.

Thank you