## 42 Citations

Supertropical Monoids: Basics, Canonical Factorization, and Lifting Ghosts to Tangibles

- Mathematics
- 2011

Supertropical monoids are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical…

Dominance and Transmissions in Supertropical Valuation Theory

- Mathematics
- 2011

This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation ϕ ≥ ψ between supervaluations ϕ and ψ on R, aiming at an enrichment of…

Commutative ν-Algebra and Supertropical Algebraic Geometry

- Mathematics
- 2019

This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over…

Scheme theoretic tropicalization

- Mathematics
- 2015

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…

Basic Operations on Supertropical Quadratic Forms

- Mathematics
- 2017

In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V ) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V ) has two free…

Minimal orderings and quadratic forms on a free module over a supertropical semiring

- Mathematics
- 2016

Supertropical Quadratic Forms II

- Mathematics
- 2015

This article is a sequel of [4], where we introduced quadratic forms on a module V over a supertropical semiring R and analysed the set of bilinear companions of a quadratic form q : V → R in case…

## References

SHOWING 1-10 OF 20 REFERENCES

Dominance and Transmissions in Supertropical Valuation Theory

- Mathematics
- 2011

This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation ϕ ≥ ψ between supervaluations ϕ and ψ on R, aiming at an enrichment of…

THE M-VALUATION SPECTRUM OF A COMMUTATIVE RING

- Mathematics
- 2002

Harrison and Vitulli introduced the notion of a V-valuation of a commutative ring, which includes the standard concepts of valuation due to Krull, Artin, Samuel and Manis (see [HV] and the…

On valuation spectra

- Mathematics
- 1998

If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic…

Non-archimedean amoebas and tropical varieties

- Mathematics
- 2004

Abstract We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For…

Supertropical matrix algebra

- Mathematics
- 2008

AbstractThe objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows:
The tropical determinant (i.e.,…

Convexity, valuations and Prüfer extensions in real algebra

- Mathematics
- 2005

We analyse the interplay between real valuations, Prufer extensions and convexity with respect to various preorderings on a given commutative ring. We study all this first in preordered rings in…

Tropical Mathematics

- Mathematics
- 2004

This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical approach in…

Supertropical matrix algebra II: Solving tropical equations

- Mathematics
- 2009

We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to…