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We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of -points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of -schemes parameterized by a moduli space of valuations on that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.
In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for . The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.
Let be a constant-degree polynomial, and let be finite sets of size . We show that vanishes on at most points of the Cartesian product , unless has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over , and a similar statement holds when have different sizes (with a more involved bound replacing ). This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.
Let be a finite extension of , and let . There is a very useful classification of -adic representations of in terms of cyclotomic -modules (cyclotomic means that where is the cyclotomic extension of ). One particularly convenient feature of the cyclotomic theory is the fact that the -module attached to any -adic representation is overconvergent.
Questions pertaining to the -adic local Langlands correspondence lead us to ask for a generalization of the theory of -modules, with the cyclotomic extension replaced by an infinitely ramified -adic Lie extension . It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely, the case of a Lubin–Tate extension, most -modules fail to be overconvergent.
In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent -modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that -modules attached to -analytic representations are overconvergent.