The tropical-arithmetic tag has no usage guidance.

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### Volume in tropical geometry as compared to volume in convex geometry

In tropical geometry, is there a notion of volume. Maybe one with some of the properties as found in classical convex geometry? If so, is there a good reference that elaborates on this question.
I ...

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### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

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### choosing between the two ways to tropicalize

When tropicalizing a subtraction-free expression (see Do all subtraction-free identities tropicalize?), is it more common to replace addition by max or by min?
Related issues:
Is there a name for ...

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### Name and notation for a binary operation

Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore ...

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### Do all subtraction-free identities tropicalize?

If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+...

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### Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...

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### Algorithms for “Ideals” in polynomial algebras over the max-plus semi-ring

I'm a beginner in tropical geometry, and I'm running into the following question:
In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether ...

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### Tropical homological algebra

Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...

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### Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...

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### Tropical mathematics and enriched category theory

Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...

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### How should one approach tropical mathematics?

Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across this paper which presented an idea that struck me as really remarkable....

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### “Wick rotation” of tropical geometry

This question is related to my earlier, even more open-ended question on tropilcalization. I will give some background and ask my question at the end.
On R, consider the family of commutative, ...