Talk:Tropical geometry

Min versus max

It's an arbitrary convention whether one works with the min-plus or the max-plus algebra, but one needs to make a choice and stick with it. The article currently chooses to use the min convention. There doesn't seem to be any point in changing over to the max convention, but perhaps there should be a brief comment somewhere in the article. Ishboyfay (talk) 04:34, 21 November 2008 (UTC)

Slapped a sentence in to that effect.

You know, this is probably the sort of article I should go and write up expandedly, in my copious free time... 4pq1injbok (talk) 05:22, 2 December 2008 (UTC)

The two algebras are isomorphic, not the same. There are certainly similarities between the two semirings but I don't think the choice of additive operation is entirely arbitrary. —Preceding unsigned comment added by 208.66.176.135 (talk) 22:24, 12 January 2010 (UTC)

People who look up one of these semi-rings may not be aware or in need of the other, or need the information that the other exists at all. I think it is helpful to have two articles —Preceding unsigned comment added by 152.1.205.239 (talk) 13:51, 5 November 2010 (UTC)

It seems that there is consensus to keep the two articles, and I would guess that everybody agrees that it would be good to have See-Also templates on each. Is that correct? Is there any objection to introducting such See-Also templates and removing the merge discussion notice? Kiefer.Wolfowitz (talk) 21:26, 16 November 2010 (UTC)

The article uses the "min" convention, but the curve in the picture follows the "max" convention. This should be fixed. For example, if someone knows how to rotate the picture by 180 degrees, that would work. Ishboyfay (talk) 19:47, 5 June 2015 (UTC)

The article declares that it uses the "min" convention, but the only actual definition in fact uses "max" — and in the very next paragraph. Yes, there is a segue stating that the definition is in the "max" convention, but this still seems very strange. Tristan (talk) 19:09, 31 January 2017 (UTC)

Reference

A new editor has objected to the reference Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 0-521-84802-4. Zbl 1133.68067. at page 211 for the definition of "tropical semiring (also known as the min-plus algebra due to the definition of the semiring)", claiming that it is "not truly related to the subject" [1]. I disagree: it seems an entirely appropriate source for that definition. I propose to reinstate it unless a better reference is suggested. Deltahedron (talk) 19:05, 3 February 2013 (UTC)

I don't have a canonical reference in mind, but indeed Lothaire is a very unusual one, for this concept. Arcfrk (talk) 22:27, 5 February 2013 (UTC)

Feel free to provide a better one then. I happen to have Lothaire to hand and the article is currently woefully short of references. Deltahedron (talk) 07:42, 6 February 2013 (UTC)

The name of the subject

The remarks in the header about the name of the subject read more like a commentary on the article rather than part of the article, especially the question about whether the name reflects the French view of Brazil and the loaded word "blames." Also, if there are going to be extensive remarks on the name (more than a single sentence), then I think those remarks belong somewhere in the body rather than in the header. Maybe it's time to create a short new section about the history of the subject. — Preceding unsigned comment added by Ishboyfay (talk • contribs) 19:57, 3 September 2014 (UTC)

Sections on applications and history

I've just started a new section on the history of the subject, although thus far it's just a couple of remarks about how the subject got its name (transplanted from the introduction). Both this section and the section on applications ought to be expanded. Ishboyfay (talk) 18:31, 20 September 2014 (UTC)

How does a tropical algebra differ from a max-plus algebra?

Why is this article separate from max-plus algebra? The difference should be explained in each of the two articles. J. Finkelstein (talk) 18:13, 8 June 2015 (UTC)

This article is about a topic in geometry, based on a construction in algebra. As far as I can see, this is already clearly stated. Ishboyfay (talk) 04:11, 14 June 2015 (UTC)

Figure has been rotated

The figure at the top of the article has been rotated, so that it now agrees with the min-plus convention of the article. Ishboyfay (talk) 03:22, 20 June 2015 (UTC)

"Tropical surface"

The section § Basic definitions states:

The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface.

Then it proceeds to provide "two important characterizations of these objects". That numbered "1." indeed says "Tropical hypersurfaces are …"; however that numbered "2." says "Tropical surfaces are …" (emphasis added). Are they talking about the same thing? Is, hypersurface, perhaps, too clumsy for conversation, so that "tropical surface" is just an informal way of saying "tropical hypersurface"? Or is there some distinct concept of "tropical surface"? If so, it's not referenced or defined in the article. yoyo (talk) 08:45, 25 July 2018 (UTC)

Valued fields examples

The section "Algebra background" currently ends with the examples:

Some common valuated fields encountered in tropical geometry are:

• $\mathbb {Q}$ or $\mathbb {C}$ with the trivial valuation, $v(a)=\emptyset$ for all $a\neq 0$.
• $\mathbb {Q}$ or its extensions with the p-adic valuation, $v(p^{n}a/b)=n$ for a and b coprime to p.
• The field of Laurent series $\mathbb {C} (\!(t)\!)$ (integer powers), or the field of (complex) Puiseux series $\mathbb {C} \{\!\{t\}\!\}$, with valuation returning the smallest exponent of t appearing in the series.

The author of these seem to have been thinking about the min convention only, which becomes confusing since the rest of that section has put equal weight on the min and max conventions; while there is some wording further up to the effect that this article applies the min convention, it is very far from obvious that this should start to apply at this particular point.

The examples are also weird in that they presume the reader knows what a valuation (algebra) is, despite it immediately before being necessary to explain the algebraic structure that the valuation maps into – a reader needing that introduction would not know that a valuation maps $0$ to infinity! Finally I don't see why one would suddenly want to use $\emptyset$ as notation for the tropical zero. Suggested rewrite:

Some common valuated fields encountered in tropical geometry, as valuated under the min convention, are:

• $\mathbb {Q}$ or $\mathbb {C}$ with the trivial valuation, $v(a)=0$ for all $a\neq 0$ and $v(0)=\infty$.
• $\mathbb {Q}$ or its extensions with the p-adic valuation, $v(p^{n}a/b)=n$ for a and b coprime to p (and $v(0)=\infty$, as always).
• The field of Laurent series $\mathbb {C} (\!(t)\!)$ (integer powers), or the field of (complex) Puiseux series $\mathbb {C} \{\!\{t\}\!\}$, with valuation returning the smallest exponent of t appearing in the series.

--78.73.97.76 (talk) 13:21, 17 July 2020 (UTC)

Removing a paragraph; let me know if there's anything worth saving here.

I'm removing a badly-spelled and -formatted paragraph that seems not to have much relevance to the actual history of the either tropical geometry itself or its name. Additionally, the sources don't appear to have anything to do with any of the claims actually made. Let me know or address it yourself if there's anything worth saving; otherwise it seems better to me simply to excise it entirely. Here's the text removed:

Quite independently of the use of the term ``tropical'' by the Brazilian school concerning the new mathematics, V.P. Maslov, in connection with the situation that occurred in the USSR during the 1980's (the ``perestroika'' years) and the opening of the ``iron curtain'' between the communist and capitalist countries, used this term repeatedly beginning from the mid 1980's. In numerous oral presentations (e.g. at the General Meetings of the USSR Academy of Sciences) and articles in the Soviet press, he warned that the opening of the curtain between the two groups of countries would inevitably lead to a situation similar to the one that arose in the 17th century when the ocean ceased to be boundary between Europe and the Americas, on the one hand, and Africa on the other. At that time cheap European and American goods were exchanged for expensive Africa ressourses  and the slave trade began. This unequal exchange could be described by the rules of the new arithmetic, which Maslov and his followers first called ``idempotent (tropical) analysis'' or ``Min; + algebra''. The term ``idempotent'' did not survive in the literature, while the term ``tropical'', being more expressive and pleasant to the ear, became more popular, although different schools gave it somewhat different meanings.

From revision https://en.wiki.x.io/w/index.php?title=Tropical_geometry&diff=prev&oldid=971396383

A Lesbian (talk) 21:07, 4 November 2020 (UTC)

References

which way around

addition is replaced with minimization and multiplication is replaced with ordinary addition

I wonder why it's that way, rather than replacing multiplication with min() and addition with itself. I was about to speculate that it has to do with identities and inverses, but there are no inverses under min()! —Tamfang (talk) 22:13, 29 June 2023 (UTC)