Krull-tropical hypersurfaces
Fuensanta Aroca (Instituto de Matem\'aticas UNAM. M\'exico)
TL;DR
This paper extends tropical geometry concepts to arbitrary ordered groups, defines tropicalization over Krull-valued fields, and generalizes Kapranov's theorem to algebraically closed fields with valuations.
Contribution
It introduces a generalized framework for tropical hypersurfaces over arbitrary ordered groups and extends key theorems to broader algebraic settings.
Findings
Extended tropical hypersurface concepts to arbitrary ordered groups
Defined tropicalization for polynomials over Krull-valued fields
Generalized Kapranov's theorem to algebraically closed fields with valuations
Abstract
The concepts of tropical-semiring and tropical hypersurface, are extended for an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator "tropicalization" we conclude with an extension of Kapranov's theorem to algebraically closed fields together with a valuation over an ordered group.
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Taxonomy
TopicsPolynomial and algebraic computation · Commutative Algebra and Its Applications · Logic, programming, and type systems