[Paper Review] Tropical Geometry over Higher Dimensional Local Fields

Shilpak Banerjee|arXiv (Cornell University)|May 30, 2011

Polynomial and algebraic computation9 references3 citations

TL;DR

This paper introduces tropicalization for closed subschemes of algebraic tori over higher-dimensional local fields, generalizing classical tropical geometry over rank-one valuation fields. It establishes that the tropicalization of a d-dimensional irreducible subscheme over an n-dimensional local field forms a rational polyhedral complex of dimension nd, with an example showing such complexes need not be pure-dimensional.

ABSTRACT

We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind, Speyer and Sturmfels to higher local fields.

Motivation & Objective

• To extend tropical geometry from rank-one valued fields to higher-dimensional local fields, which have vector-valued valuations with lexicographic ordering.
• To define and study the tropicalization of closed, reduced, irreducible subschemes of algebraic tori over n-dimensional local fields.
• To generalize the classical result that tropicalizations are pure-dimensional rational polyhedral complexes to the higher-rank setting.
• To investigate whether tropicalizations over higher local fields retain purity, and to construct counterexamples if not.

Proposed method

• Define a vector-valued valuation ν: (K^al)^× → Γ_R ≅ ℝ^n with lexicographic order, generalizing the rank-one valuation in classical tropical geometry.
• Construct the tropicalization map Trop as a dual to the coordinate-wise valuation, mapping points of the torus to Hom(M, Γ_R), where M is the character lattice.
• Use retraction maps r_j^n: Trop^{(n)}(X) → Trop^{(j)}(X) to inductively analyze the dimension of tropicalizations over j-dimensional local fields.
• Introduce the n-extended Newton polytope for hypersurfaces to provide a geometric proof of the dimension formula.
• Analyze initial degenerations of polynomials under various weight conditions to compute tropicalizations via degeneration tables.
• Apply the theory of convex subgroups and ordered abelian groups to handle the structure of higher-rank value groups.

Experimental results

Research questions

• RQ1 What is the correct generalization of tropicalization for subschemes of tori over higher-dimensional local fields?
• RQ2 What is the dimension of the tropicalization of a d-dimensional closed, reduced, irreducible subscheme over an n-dimensional local field?
• RQ3 Does the tropicalization over higher local fields remain a pure-dimensional rational polyhedral complex, as in the classical case?
• RQ4 Can one construct explicit examples where the tropicalization fails to be pure-dimensional?
• RQ5 How do initial degenerations and weight conditions determine the structure of the tropicalization in higher-rank settings?

Key findings

• The tropicalization of a d-dimensional closed, reduced, irreducible subscheme of an algebraic torus over an n-dimensional local field is a rational polyhedral complex of dimension nd.
• For hypersurfaces, the dimension result is proven independently using the n-extended Newton polytope, generalizing the construction of Einslieder, Kapranov, and Lind.
• An explicit example is constructed where the tropicalization of a hypersurface over ℂ((t₁))((t₂)) is not pure-dimensional, with a one-dimensional maximal face joining (1,0,0.5,0) to (2,0,1,0).
• The tropicalization map is not necessarily pure-dimensional in higher local fields, contradicting the classical pure-dimensionality result.
• The value group Γ_R ≅ ℝ^n carries a lexicographic order, which introduces asymmetry and complicates the extension of the valuation to the algebraic closure.
• The theory of higher local fields allows for a natural generalization of tropical geometry, with potential applications to analytification and Gröbner basis theory over higher-rank valuations.