Tropical geometry: Difference between revisions

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(→‎Applications: improved wording)
: <math>x \otimes y = x + y.</math>
So for example, the classical polynomial <math>x^3 + 2xy + y^4</math> would become <math>\min(\{x+x+x,\; 2+x+y,\; y+y+y+y)\}</math>. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains.
Tropical geometry is a variant of [[algebraic geometry]] in which polynomial graphs resemble [[piecewise linear manifold|piecewise linear]] meshes, and in which numbers belong to the [[tropical semiring]] instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic geometry, such as the [[Brill–Noether theorem]], using the tools of tropical geometry.<ref>{{Cite web|url=|title=Tinkertoy Models Produce New Geometric Insights|last=Hartnett|first=Kevin|website=[[Quanta Magazine]]|access-date=2018-12-12}}</ref>
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