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Please allow me to ask a potentially dumb question (or maybe more precisely, a question floating on clouds of ignorance):

Why is a max-plus algebra called a tropical algebra?

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tropical refers to Brazil. Imre Simon working in Brazil was a pioneer of the field. –  Colin Tan Sep 23 '11 at 8:30
    
I see! But surely max-plus was studied even before Imre, or not? –  Suvrit Sep 23 '11 at 8:33
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He was working at Sa\~o Paulo, I believe, and I think that this city is a little bit below the Capricorn tropic!! –  Fernando Muro Sep 23 '11 at 8:39
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Warning: some Japanese authors use "tropicalization" to refer to lifting a max-plus formula to a rational function, which is what most Western authors call "detropicalization". I think the Japanese point of view makes better sense, in that a max-plus formula is like a zero Kelvin version, like Kashiwara's crystal basis. But it seems unlikely to be uprooted now. –  Allen Knutson Sep 24 '11 at 1:46
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2 Answers

up vote 17 down vote accepted

A lot of sources mention that the adjective "tropical" is given in honor of Imre Simon, but it seems hard to find who precisely coined the term. I found some sources which attribute this to some French mathematicians. Here is what Bryan Hayes writes on the topic:

For starters, what is that word “tropical” supposed to mean? Speyer and Sturmfels explain: “The adjective tropical was coined by French mathematicians, including Jean-Eric Pin, in honor of their Brazilian colleague Imre Simon.” Pin, in a 1998 paper (.pdf), deflects the credit to another French mathematician, Dominique Perrin, again noting that the name honors “the pioneering work of our brazilian colleague and friend Imre Simon.” Simon himself, in a 1988 paper (.ps), attributes the term to yet a third French mathematician, Christian Choffrut. Apparently, no one wants to lay claim to the word, and I can’t entirely blame them. Speyer and Sturmfels go on: “There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.”

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Thanks. This settles it for me :-) –  Suvrit Sep 23 '11 at 11:59
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From http://en.wikipedia.org/wiki/Tropical_geometry:

The adjective tropical is given in honor of the Brazilian mathematician Imre Simon, who pioneered the field.

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Grazie! I should have read the article on "tropical geometry"; but it seems that max-plus had been around even before Imre? –  Suvrit Sep 23 '11 at 8:34
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Given the name of the guy, why is it not called ``Balaton Geometry''? –  Igor Rivin Sep 23 '11 at 8:37
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A detailed history of the subject is given in: arxiv.org/PS_cache/math/pdf/0507/0507014v1.pdf According to this survey, the first example of the kind was given by S.C. Kleene (1956), while a systematic treatment of idempotent linear algebra and related subjects was initiated by V.P. Maslov in 1980s. However his approach was analytic, not geometrical, thus the tropical geometry was rediscovered a decade later. –  Anatoly Kochubei Sep 23 '11 at 13:29
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