On the dimension of polynomial semirings

Joó, Dániel; Mincheva, Kalina

doi:10.1016/j.jalgebra.2018.04.007

Cited by 7 publications

(1 citation statement)

References 9 publications

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“…Second, we take the first steps towards algebraic geometry over F ∞ . Our theory is mainly motivated by a novel approach by Dániel Joó and Kalina Mincheva ( [5], [6]). One of the key ideas in these papers is that congruences are more natural objects to study than ideals.…”

Section: Introductionmentioning

confidence: 99%

Algebraic geometry over the residue field of the infinite place

Hablicsek

Juhász

2017

Communications in Algebra

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Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, F∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.Proof. These are all trivial consequences of theorems in universal algebra.Proposition 2.11. Hom(M 1 , M 2 ) has a natural F-structure.Proof. The pointwise sum and scalar multiple of two homomorphisms is a homomorphism.Definition 2.12. Given two F-modules M 1 and M 2 , the tensor product M 1 ⊗ M 2 is a module with a natural bilinear map M 1 ×M 2 → M 1 ⊗M 2 where M 1 ×M 2 is the set of pairs, such that for any bilinear map M 1 × M 2 → N for some other module N , there is exactly one map M 1 ⊗ M 2 → N that makes the diagramProposition 2.13. Given two F-modules M 1 and M 2 , M 1 ⊗ M 2 exists and is unique. It is generated by elements of the form m 1 ⊗ m 2 with m 1 ∈ M 1 and m 2 ∈ M 2 . Furthermore · ⊗ M is a covariant functor.

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Section: Introductionmentioning

confidence: 99%

Algebraic geometry over the residue field of the infinite place

Hablicsek

Juhász

2017

Communications in Algebra

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A general framework for tropical differential equations

Giansiracusa

Mereta

2023

manuscripta math.

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We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev’s framework for tropical differential equations. We then prove a differential analogue of Payne’s inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

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Tropical adic spaces I: the continuous spectrum of a topological semiring

Friedenberg,

Mincheva

2024

Res Math Sci

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Toward building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow-up paper.

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On the dimension of polynomial semirings

Cited by 7 publications

References 9 publications

Algebraic geometry over the residue field of the infinite place

Algebraic geometry over the residue field of the infinite place

A general framework for tropical differential equations

Tropical adic spaces I: the continuous spectrum of a topological semiring

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