# Questions tagged [tropical-arithmetic]

The tropical-arithmetic tag has no usage guidance.

26
questions

2
votes

0
answers

61
views

### Does tropicalization commute with composition?

Say we have two polynomials
$$
f = \sum d_n x^n, \quad g = \sum_n b_n x^n
$$
that tropicalize to
$$
F = \max(d_n + nx), \quad G = \max(b_n + nx).
$$
Can we say $f\circ g$ tropicalizes to $F\circ G$?
...

1
vote

0
answers

23
views

### Proving equivalence of tropical polynomials

I am new to the world of tropical mathematics. I am wondering if there is an algorithm to prove the equivalence of two tropical polynomials (in the plus-min semiring let's say), say over multivaribles?...

4
votes

0
answers

166
views

### Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...

0
votes

0
answers

52
views

### A system of inequalities involving a skew-symmetric integer matrix

Which skew-symmetric integer matrices $S$ satisfy the following inequalities
$SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$
where
$V_i$ denotes the column with integer entries such that the $i$-th ...

1
vote

0
answers

71
views

### Tropical Arithmetic and Numeral Systems - Number systems [closed]

Is there some paper about Numeral Systems / Number Systems, using tools of Tropical Geometry to represent numbers? Maybe through Continuous fractions, triangular numbers, arithmetic functions, ...

4
votes

0
answers

127
views

### Tropical lie algebra

In this article https://arxiv.org/pdf/1705.01075.pdf are we mean that Lie semialgebras over semirings with a negation map is tropical version of Lie algebra?. And what we do when we define lifting? ...

2
votes

0
answers

78
views

### Tropical algebraic structures

What is the difference between tropical lie semialgebra and lie semialgebra with anegation map? and How can I build another algebraic structure in tropical algebra?

3
votes

0
answers

275
views

### Factorization of tropical polynomials

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here a tropical polynomial ...

2
votes

1
answer

255
views

### Can we have "tropical polynomials" with arbitrary real powers?

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a ...

2
votes

1
answer

962
views

### Is there any "fundamental" distinction between min-plus, max-plus, min-product, and max-product algebras?

In the paper Faster Algorithms for Max-Product Message Passing by McAuley and Caetano (see e.g. here or here), several statements are made which seem mathematically questionable to me.
For ...

3
votes

0
answers

113
views

### Max-Plus algebra and hyperplane arrangements

Given an expression in the Max-Plus algebra is it possible to recognize if it represents a continuous piecewise linear (CPWL) function whose polyhedral complex is a hyperplane arrangement?
Or ...

3
votes

0
answers

203
views

### Can MacLane's notion of universality inform $\mathbb{F}_1$?

MacLane (1939) calls a field $F$ universal if every other field $F'$ of the same cardinality and characteristic as $F$ is a subfield of $F$. He then exhibits an example, viz. a field of generalized ...

4
votes

1
answer

173
views

### Does a nontrivial notion of integral under logarithmic deformations of $\mathbb{R}_+$ exist?

Background
The upper and lower Maslov dequantizations are respectively the limits $h \downarrow 0$ and $h \uparrow 0$ of deformations of the semifield $(\mathbb{R}_+,+,\cdot)$ defined for $0 \ne h \...

6
votes

1
answer

783
views

### Is there any structure theorem for piecewise linear functions?

I was wondering if such statements are known like "any piecewise linear function from $\mathbb{R}^d \rightarrow \mathbb{R}$ can be written as $\sum_{i=1}^k \alpha_i (\text{ some $2$ piece linear ...

0
votes

1
answer

226
views

### 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 ...

9
votes

1
answer

638
views

### 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 ...

3
votes

0
answers

443
views

### 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 ...

5
votes

2
answers

425
views

### 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 ...

19
votes

2
answers

701
views

### 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+...

10
votes

3
answers

1k
views

### 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 ...

2
votes

0
answers

159
views

### 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 ...

18
votes

1
answer

2k
views

### 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 ...

5
votes

0
answers

494
views

### 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 ...

16
votes

1
answer

1k
views

### 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 ...

39
votes

5
answers

4k
views

### 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 Litvinov - The Maslov dequantization, idempotent and tropical ...

4
votes

1
answer

854
views

### "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, ...