Comments on: Semiring analogies

By: Ars Mathematica » Blog Archive » Tropical Geometry

Ars Mathematica » Blog Archive » Tropical Geometry — Thu, 12 Jan 2006 04:16:42 +0000

[...] We’ve discussed semirings before. One interesting application is tropical geometry, which studies the analogue of algebraic varieties over the max-plus semiring (sometimes known as the tropical semiring). Grigory Mikhalkin has posted a survey article on the subject, Tropical geometry and its applications, to arXiv. (The “applications” of the title are applications to ordinary algebraic geometry.) [...]

By: Walt

Walt — Sun, 16 Oct 2005 04:55:28 +0000

Oops. I initially started with max-plus, but when I saw that sigfpe was min-plus, I changed it (but apparently not quite everywhere). Now fixed.

By: ansobol

ansobol — Mon, 10 Oct 2005 19:38:42 +0000

Strictly speaking, in the third example you have to take infinite _infima_, not suprema; you have also to add a minus infinity to ensure they exist (or restrict to bounded sets). With the latter provision (summing only bounded infinite subsets) all three examples you give enjoy the “easy summability” property. This observation has been systematically developed by the French school (where sigfpe’s knowledge comes from) and by Russians including [Grigory Litvinov](http://arxiv.org/abs/math/0009128).

The subject is indeed fascinating, and therefore it has been rediscovered several times under different names. The most recent incarnation is called “tropical math” and deals mostly with algebraic geometry; the idea is basically that max-plus algebraic geometry deals with convex polygones instead of algebraic curves. There is a little known [arXiv article](http://arxiv.org/abs/math/0005163) by Oleg Viro, which makes for a very good and readable introduction into the “tropical algebraic geometry.” More recent literature can easily be found in arXiv if you look for keywords such as “tropical” and “amoeba”; I also keep a bibliography [at CiteULike](http://www.citeulike.org/user/ansobol/tag/tropical); sorry for the plug

A very complete bibliography up to the late 1990s exists at [Stéphane Gaubert's site](http://amadeus.inria.fr/gaubert/PAPERS/abstract/abstract.html), but it does not cover the most recent “tropical” development.

By: sigfpe

sigfpe — Mon, 10 Oct 2005 03:44:43 +0000

By the way, John Baez occasionally talks about semirings in This Week’s Finds but it doesn’t show up in web searches as he calls them rigs. The Classical/Quantum analogy came from there but I can’t find the exact link again, even with Google.

Anyway…I’m beginning to get obsessed with semirings myself…