Tropical analytic geometry, Newton polygons, and tropical intersections

Abstract

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 , . . . , fn 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 , . . . , fn. 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. 1 1.4. From the perspective of a tropical geometer, the theory of rigid spaces is useful because the analytic topology on R n is much better approximated by the rigid-analytic topology on the torus G n m . For example, the unit box [0, 1] n is an analytic neighborhood in the Euclidean space R n , yet trop −1 ([0, 1] n ) ⊂ |G n m | is the n-fold product of the annulus {ξ ∈ K × : val(ξ) ∈ [0, 1]}, which is a very nicely behaved rigid-analytic object (it is a smooth affinoid space), but is not the set of points underlying a subscheme. Similarly, (R ∪ {∞}) n can naturally be regarded as the tropicalization of the affine space A n (see §5), under which identification the unit ball B n is the inverse image under trop of (R ≥0 ∪ {∞}) n (a neighborhood of the point (∞, . . . , ∞)). 1.5.The following example is an application of rigid-analytic methods to a tropically-local problem. Let U {0} = trop −1 ({0}) ⊂ |G n m |. This is an affinoid space, which implies (see §4) that it is the maximal spectrum of the algebrais the associated subscheme then |Y | ∩ U {0} is identified with the set of maximal ideals of K U {0} containing the ideal aK U {0} , so to show that 0 ∈ Trop(Y ) is equivalent to showing that a does not generate the unit ideal in K U {0} . This ends up being equivalent to the well-known criterion that the initial ideal of a at 0 contain no monomials. The characterization of the tropicalization (or rather the Bieri-Groves set) of a scheme by initial ideals was proved by Einsiedler-Kapranov-Lind [EKL06] using these methods; we give a treatment below (7.9) which also applies to tropicalizations of analytic spaces. (The first complete proof of this theorem was given by Draisma [Dra08, Theorem 4.2] and also uses affinoid algebras, albeit in a different way; see 7.13.)1.6. A family of translations of a tropical variety parameterized by an interval corresponds to a family of subvarieties of a torus parametrized by a rigid-analytic annulus. We study such families in order to prove the theorem indicated in (1.1,ii); to illustrate the main idea we will sketch a special case., and suppose that Trop(f 1 ) ∩ ...

On the structure of non-archimedean analytic curves

“…This is a polytopal domain in G an m [Gub07,Rab12]; it is therefore an affinoid space whose ring of analytic functions is…”

“…This is a polyhedral domain in the sense of [Rab12]; more precisely, it is the affinoid domain with ring of analytic functions…”

Constructing polynomial systems with many positive solutions using tropical geometry

“…second) equation of (4.4). E. Brugallé and L. López De Medrano showed in [5, Proposition 3.11] (see also [11,17,18] for more details for higher dimension and more exposition relating toric varieties and tropical intersection theory) that the number of solutions of (4.2) with valuation at v 0 is equal to the mixed volume MV( ξ 1 , ξ 3 ) of ξ 1 and ξ 3 (recall that v 0 = ξ 1 + ξ 3 ). Since we assumed that (4.2) has only non-degenerate solutions in (K * ) 2 , we get MV( ξ 1 , ξ 3 ) distinct solutions of the system (4.2) in (K * ) 2 ( ξ 1 , ξ 3 ).…”

“…Then, to each such ξ , we associate a certain polynomial sub-system of (1.4), called reduced real systems with respect to ξ (see [8,Chapter 2.2.6]). All together, those reduced systems approximate all the non-degenerate parametrized solutions to (1.4) (see [11,17,18] and [5]). We adapt this approach to our setting by considering a particular type of parametrized non-degenerate solutions (α 1 (t), α 2 (t)), which we also call positive (i.e.…”

Tropical Skeletons

“…We briefly introduce these partial compactifications here; see [31,33] for details. Put R = R∪{∞}, and for a rational pointed cone σ ⊂ N R we let N σ R denote the space of monoid homomorphisms Hom(S σ , R).…”