Fibers of tropicalization
By Sam Payne. 2007
88 citations recorded by the source corpus.
We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained “constructively” using Kedlaya’s transfinite version of Newton’s method.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/ad17ae4bac43736dd550b99d4cb52bcbe1c838d0
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In Algebra and Number Theory
In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Analytification is the limit of all tropicalizations (2008, background): “The equivalence of (2)–(5) is standard; see [SS04a] and [Pay07].”
- Linear series on metrized complexes of algebraic curves (2012, background): “As noted by Payne in [41], exploded tropicalizations can be thought of as an algebraic analogue of the “exploded torus fibrations” studied by Parker from a symplectic viewpoint [40].”
- Equations for Chow and Hilbert quotients (2007, background): “u 1;:::;u m) 2V K(J)g: This equals the set fw2Rm: in w(I) 6=h1ig: There is a polyhedral fan whose support is Trop(Y). Versions of this result appear in [EKL06], [SS04, Theorem 2.1], [Dra08], [JMM08], [Pay09]. We consider here only the \constant coecient" case, where the coecients of polynomials”
- A Tropical Toolkit (2006, methodology): “provided an algorithm that finds a tropical lift in [19]. This algorithm uses some ideas from our proof and their paper is recommended as an exposition of our proof in terms of commutative algebra. In [29], by applying a projection argument to reduce to the hypersurface case, Payne gave a stronger ve”
- A Field of Generalised Puiseux Series for Tropical Geometry (2007, unknown): “We define the t-support of F as the set t-supp(F ) = {( i, val(fi) ) | i = 0, . . . , n, fi 6= 0 } ⊂ R2, and we call the convex hull, say N(F ), of t-supp(F ) the t-Newton polygon of F .”