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| {{Use dmy dates|date=August 2013}}
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| [[File:Cubique tropicale.svg|thumbnail|A tropical cubic curve]]
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| '''Tropical geometry''' is a relatively new area in [[mathematics]], which might loosely be described as a [[piecewise linear manifold|piece-wise linear]] or skeletonized version of [[algebraic geometry]]. Its leading ideas had appeared in different guises in previous works of George M. Bergman and of Robert Bieri and John Groves, but only since the late nineties has an effort been made to consolidate the basic definitions of the theory. This effort has been in great part motivated by the strong applications to [[enumerative geometry|enumerative algebraic geometry]] uncovered by [[Grigory Mikhalkin]].
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| The adjective ''[[tropical]]'' is given in honor of the [[Brazil]]ian mathematician [[Imre Simon]], who pioneered the field.
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| == Basic definitions ==
| | Before you invest loads of funds on things like controls and for memory cards, appear the net for a secondhand variation. Occasionally a store will probably feel out of used-game hardware, which could be very affordable. Make sure you look with just one web-based seller's feedback prior to making the purchase so a few seconds . whether you are having what you covered.<br><br>Video games are fun to explore your kids. When you have just about any queries relating to where in addition to how you can work with [http://prometeu.net clash of clans hack tool android], you are able to email us at our site. This can help you learn much more info on your kid's interests. Sharing interests with your kids like this can likewise create great conversations. It also gives an opportunity to monitor expansion of their skills.<br><br>Take pleasure in unlimited points, resources, coinage or gems, you needs to download the clash of clans crack tool by clicking to your button. Depending regarding the operating system that tend to be using, you will need to run the downloaded list as administrator. [http://www.Google.de/search?q=Provide Provide] the log in ID and judge the device. Proper this, you are would be smart to enter the number of gems or coins that you would like to get.<br><br>Computer games offer entertaining that would everybody, and they could be surely more complicated than Frogger was! And get all you have the ability to out of game titles, use the advice laid out here. An individual might be going to find each exciting new world into gaming, and you will probably wonder how you ahead of got by without the company!<br><br>A blog is offering Clash of Clans hack tool personal trainer to users who are interested. The website offering this tool remains safe and secure and it guarantees greatest software. There will also other sites which secure the tool. But take into account either incomplete or to do with bad quality. when users download these partially hack tools, instead of doing well they end up in trouble. So, players are advised to select the tool from a website that offers complete companies.Users who are finding it tough to fold the hurdles can locate a site that allows folks to download the cheats. Most of the businesses allow free download and websites charge fees. Users can locate a niche site from where they acquire good quality software.<br><br>It's tough to select the appropriate xbox game gaming console. In the beginning, you should involving your standard requirements as the video game player, as well as check out the extras made available from unit you are since. Consider investigating on-line. Check consumer reviews to ascertain if more gamers have discovered difficulty with the unit. For you to buying a game process, you should know up to a whopping you are able to help regarding it.<br><br>You actually are playing a flaunting activity, and you too don't possess knowledge including it, establish the ailment stage to rookie. This should help you may pick-up in the awesome options that come although game and discover in your direction round the field. Should you set that it more than that, you will likely get frustrated and indicates possess fun. |
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| ''We will use the ''min convention'', that tropical addition is classical minimum. It is also possible to cast the whole subject in terms of the ''max convention'', negating throughout, and several authors make this choice.''
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| Consider the ''tropical [[semiring]]'' (also known as the ''min-plus [[algebra]]'' due to the definition of the semiring). This semiring, (ℝ ∪ {∞}, ⊕, ⊗), is defined with the operations as follows:
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| : <math> x \oplus y = \min\{\, x, y \,\},\,</math>
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| : <math> x \otimes y = x + y.\,</math> | |
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| From this we can also define tropical exponentiation in the usual way as iterated tropical products.
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| A monomial of variables in this semiring is a [[linear map]], represented in classical arithmetic as a linear function of the variables with integer coefficients.<ref name=Speyer2004>{{cite web|last=David Speyer and Bernd Sturmfels|title=Tropical Mathematics|url=http://arxiv.org/pdf/math/0408099v1.pdf|publisher=Arxiv|accessdate=30 January 2014}}</ref> A polynomial in the semiring is the minimum of a finite number of such monomials, and is therefore a concave, continuous, piecewise linear function.
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| The set of points where a tropical polynomial ''F'' is non-differentiable is called its associated '''tropical hypersurface'''.
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| There are two important characterizations of these objects:
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| # Tropical hypersurfaces are exactly the rational polyhedral complexes satisfying a "zero-tension" condition.
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| # Tropical surfaces are exactly the non-Archimedean [[Amoeba (mathematics)|amoebas]] over an [[algebraically closed field|algebraically closed]] [[non-archimedean field]] ''K''.<ref>{{cite book | last=Mikhalkin | first=Grigory | chapter=Amoebas of algebraic varieties and tropical geometry | editor1-last=Donaldson | editor1-first=Simon | editor1-link=Simon Donaldson | editor2-first=Yakov | editor2-last=Eliashberg | editor2-link=Yakov Eliashberg | editor3-first=Mikhael | editor3-last=Gromov | editor3-link=Mikhail Leonidovich Gromov | title=Different faces of geometry | location=New York, NY | publisher=Kluwer Academic/Plenum Publishers | series=International Mathematical Series (New York) | volume=3 | pages=257-300 | year=2004 | isbn=0-306-48657-1 | zbl=1072.14013 }}</ref>
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| These two characterizations provide a "dictionary" between combinatorics and algebra. Such a dictionary can be used to take an algebraic problem and solve its easier combinatorial counterpart instead.
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| The tropical hypersurface can be generalized to a '''tropical variety''' by taking the non-archimedean amoeba of ideals ''I'' in ''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] instead of polynomials. It has been proved that the tropical variety of an ideal ''I'' equals the intersection of the tropical hypersurfaces associated to every polynomial in ''I''. This intersection can be chosen to be finite.
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| There are a number of articles and surveys on tropical geometry. The study of tropical curves (tropical hypersurfaces in ℝ<sup>2</sup>) is particularly well developed. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., [[Pappus's theorem]], [[Bézout's theorem]], the [[degree-genus formula]], and the [[elliptic curve|group law of the cubics]]<ref>{{cite book | last1=Chan | first1=Melody | last2=Sturmfels | first2=Bernd | chapter=Elliptic curves in honeycomb form | zbl=06241528 | editor1-last=Brugallé | editor1-first=Erwan | title=Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011 | location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-0-8218-9146-9 | series=Contemporary Mathematics | volume=589 | pages=87-107 | year=2013 }}</ref> all have tropical counterparts.
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| ==See also==
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| * [[Max-plus algebra]]
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| ==References==
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| {{reflist}}
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| {{Refbegin}}
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| *{{cite arXiv |last=Bogart |first=Tristram |last2=Jensen |first2=Anders |last3=Speyer |first3=David |last4=Sturmfels |first4=Bernd |last5=Thomas |first5=Rehka |authorlink= |eprint=math/0507563 |title=Computing Tropical Varieties |class=math.AG |year=2005 |version=v1 |accessdate=7 February 2013 }}
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| *{{cite arXiv |last=Einsiedler |first=Manfred |last2=Kapranov |first2=Mikhail |last3=Lind |first3=Douglas |authorlink= |eprint=math/0408311 |title=Non-archimedean amoebas and tropical varieties |class=math.AG |year=2005 |version=v2 |accessdate=7 February 2013 }}
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| *{{cite arXiv |last= Gathmann |first= Andreas |authorlink= |eprint=math/0601322 |title=Tropical algebraic geometry |class=math.AG |year= 2006|version=v1 |accessdate=7 February 2013 }}
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| *{{cite book|last=Gross|first=Mark|title=Tropical geometry and mirror symmetry|year=2010|publisher=Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation|location=Providence, R.I.|isbn=9780821852323}}
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| *{{cite book|last=Itenberg|first=Illia|coauthors=Grigory Mikhalkin, Eugenii Shustin |title=Tropical algebraic geometry|year=2009|publisher=Birkhäuser Basel|location=Basel|isbn=9783034600484|edition=2nd }}
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| *{{cite arXiv |last= Mikhalkin |first= Grigory |authorlink= |eprint=math/0601041 |title=Tropical Geometry and its applications |class=math.AG |year=2006 |version= v2|accessdate=7 February 2013 }}
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| *{{cite arXiv |last= Mikhalkin |first=Grigory |authorlink= |eprint=math/0312530 |title=Enumerative tropical algebraic geometry in R2 |class=math.AG |year=2004 |version= v4|accessdate=7 February 2013 }}
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| *{{cite arXiv |last= Mikhalkin |first=Grigory |authorlink= |eprint=math/0403015 |title=Amoebas of algebraic varieties and tropical geometry |class=math.AG |year=2004 |version= v1|accessdate=7 February 2013 }}
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| *{{cite journal|last=Pachter|first=L. |last2=Sturmfels |first2=Bernd |title=Tropical geometry of statistical models|journal=Proceedings of the National Academy of Sciences|year=2004|volume=101|issue=46|pages=16132–16137|doi=10.1073/pnas.0406010101}}
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| *{{cite arXiv |last= Speyer |first=David E. |authorlink= |eprint=math/0304218 |title=The Tropical Grassmannian |class=math.AG |year=2003 |version=v3 |accessdate=7 February 2013 }}
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| *{{cite arXiv |last=Speyer |first=David |last2=Sturmfels |first2=Bernd |authorlink= |eprint=math/0408099 |title=Tropical Mathematics |class=math.AG |year= 2004|version= v1|accessdate=7 February 2013 }}
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| *{{cite arXiv |last= Theobald |first= Thorsten |authorlink= |eprint=math/0306366 |title=First steps in tropical geometry |class=math.AG |year=2003 |version=v2 |accessdate=7 February 2013 }}
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| {{Refend}}
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| ==External links==
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| * [http://www.archive.org/movies/details-db.php?id=4603 Tropical Geometry, I]
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| [[Category:Algebraic geometry]]
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