Let @math be a Laurent polynomial in @math variables, and let @math be a generic smooth fiber of @math . In RSTZ Ruddat-Sibilla-Treumann-Zaslow give a combinatorial recipe for a skeleton for @math . In this paper, we show that for a suitable exact symplectic structure on @math , the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.
In Gammage-Shende @cite_3 , as one ingredient in proving HMS for the toric boundary of a toric variety, they constructed the Liouville skeleton for the same hypersurface as considered here. However, their results [Theorem 3.4.2] GS depends on the following hypothesis that, there exists some tropicalization function @math and some identification @math , such that the tropical amoeba polytope @math contains @math as an interior point, and @math restricts to each face @math of @math has a minimum in the interior of @math . This hypothesis is true in two-dimension, and can be verified in certain examples, e.g. mirror to weighted projective spaces. But in general, one does not know if it is always true, it would be interesting to find a proof or construct a counter-example. We thank Gammage and Shende for this clarification. Our approach here does not rely on this hypothesis, which is equivalent to " @math is adapted to the tropical amoeba polytope @math " for some choice of @math . We avoid this by considering a more flexible choice of Kahler potentials on than @math , and our approach works for any choice of @math compatible with @math .
@cite_3
2094372703
The amoeba of a holomorphic function @math is, by definition, the image in @math of the zero locus of @math under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function @math with the property that its Monge-Ampere mass is concentrated to the amoeba of @math We obtain results of two kinds; by approximating @math with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of @math ; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in @math . We also consider systems of functions @math and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.