Newton polytopes and tropical geometry

Abstract

The practice of bringing together the concepts of ‘Newton polytopes’, ‘toric varieties’, ‘tropical geometry’, and ‘Gröbner bases’ has led to the formation of stable and mutually beneficial connections between algebraic geometry and convex geometry. This survey is devoted to the current state of the area of mathematics that describes the interaction and applications of these concepts. Bibliography: 68 titles.

Cohomology rings of quasitoric bundles

“…By Theorem 5.8 (see also [KKE21] for similar results), the kernel of the graded epimorphism of algebras π : Sym(V ) → A is given by the ideal just defined: ker(π) = I ℓ , and hence A ≃ Sym(V )/I ℓ .…”

Hypergeometric Feynman Integrals

“…Amoebas as well as coamoebas provide a link between algebraic geometry and tropical geometry [180]. We suggest [147] for a recent survey about tropical geometry and its connection to toric varieties. The relations between coamoebas and Euler-Mellin integrals were investigated in [18,178], which state the key references for this section.…”

On Newton polytopes of Lagrangian augmentations

“…The same counts of such monomials for the Lagrangians associated to $$\vartheta$$‐loops are more challenging and, even when computable, typically not able to show that the $$\vartheta$$‐orbits are entire. Newton polytopes associated to multivariable Laurent polynomials are often more robust invariants than the precise count of monomials and contain a significant amount of geometric information, see, for example, [2, 24, 27–29, 34]. Through a variety of explicit computations and applications, this note will illustrate that Newton polytopes are useful in the study of the Legendrian contact dg‐algebra.…”