Tropical analytic geometry, Newton polygons, and tropical intersections

By Joseph Rabinoff. 2010

85 citations recorded by the source corpus.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/50b76d4a50f3c97c9d7624559d8817e6766683f6

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Citations in the corpus

On the structure of non-archimedean analytic curves (2013, background): “This is a polyhedral domain in the sense of [23]; more precisely, it is the affinoid domain with ring of analytic functions”
On the structure of nonarchimedean analytic curves (2014, methodology): “Gan m = trop −1(R). We use trop to deﬁne several analytic domains in A1,an: • For a ∈K× the standard closed ball of radius |a|is B(a) = trop−1([val(a),∞]). This is a polyhedral domain in the sense of [Rab12]; more precisely, it is the afﬁnoid domain with ring of analytic functions Kha−1ti= ( X∞ n=0 ”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “F. If F=SpecPis afﬁne, we can formally describe this stratiﬁcation following [Pay09, Section 3] or [Rab12, Proposition 3.4]. Write σfor the cone Hom(P,R≥0)as well as NR(σ)=Hom(P,R). Proposition 3.12 ([Rab12] Proposition 3.4). There is a natural stratiﬁcation G τ≺σ NR/Specτ −∼→ N R(σ) ofNR(σ)bylocall”
Nonarchimedean geometry, tropicalization, and metrics on curves (2011, background): “one σ= σP is pointed. Deﬁnition4.5. ThepolyhedraldomainassociatedtoPistheinverseimage undertrop : Yan σ → NR(σ) of the closure of Pin NR(σ) and is denoted U P. These polyhedral domains, introduced in [Rab12], directly generalize the polytopal domains studied by Gubler in [Gub07b]. Indeed, a polytopa”