784 research outputs found
Scheme theoretic tropicalization
In this paper, we introduce ordered blueprints and ordered blue schemes,
which serve as a common language for the different approaches to
tropicalizations and which enhances tropical varieties with a schematic
structure. As an abstract concept, we consider a tropicalization as a moduli
problem about extensions of a given valuation between ordered
blueprints and . If is idempotent, then we show that a
generalization of the Giansiracusa bend relation leads to a representing object
for the tropicalization, and that it has yet another interpretation in terms of
a base change along . We call such a representing object a scheme theoretic
tropicalization.
This theory recovers and improves other approaches to tropicalizations as we
explain with care in the second part of this text.
The Berkovich analytification and the Kajiwara-Payne tropicalization appear
as rational point sets of a scheme theoretic tropicalization. The same holds
true for its generalization by Foster and Ranganathan to higher rank
valuations.
The scheme theoretic Giansiracusa tropicalization can be recovered from the
scheme theoretic tropicalizations in our sense. We obtain an improvement due to
the resulting blueprint structure, which is sufficient to remember the
Maclagan-Rinc\'on weights.
The Macpherson analytification has an interpretation in terms of a scheme
theoretic tropicalization, and we give an alternative approach to Macpherson's
construction of tropicalizations.
The Thuillier analytification and the Ulirsch tropicalization are rational
point sets of a scheme theoretic tropicalization. Our approach yields a
generalization to any, possibly nontrivial, valuation with
idempotent and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the
paper, please cf. the paragraph "Differences to previous versions" in the
introductio
Duality and separation theorems in idempotent semimodules
We consider subsemimodules and convex subsets of semimodules over semirings
with an idempotent addition. We introduce a nonlinear projection on
subsemimodules: the projection of a point is the maximal approximation from
below of the point in the subsemimodule. We use this projection to separate a
point from a convex set. We also show that the projection minimizes the
analogue of Hilbert's projective metric. We develop more generally a theory of
dual pairs for idempotent semimodules. We obtain as a corollary duality results
between the row and column spaces of matrices with entries in idempotent
semirings. We illustrate the results by showing polyhedra and half-spaces over
the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2
- β¦