In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
So for example, the classical polynomial would become . 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 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.[1]
^Hartnett, Kevin (5 September 2018). "Tinkertoy Models Produce New Geometric Insights". Quanta Magazine. Retrieved 12 December 2018.
In mathematics, tropicalgeometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication...
Introduction to TropicalGeometry is a book on tropicalgeometry, by Diane Maclagan and Bernd Sturmfels. It was published by the American Mathematical...
respectively. The tropical semiring has various applications (see tropical analysis), and forms the basis of tropicalgeometry. The name tropical is a reference...
Versions of a tropicalgeometry, of an absolute geometry over a field with one element and an algebraic analogue of Arakelov geometry were realized in...
is cryptography based on the tropical semiring. Tropicalgeometry is an analog to algebraic geometry, using the tropical semiring. Litvinov, G. L. (2005)...
mathematics are tropicalgeometry, commutative algebra, and optimization. Nonlinear algebra is closely related to algebraic geometry, where the main objects...
algebraic geometry. Versions of a tropicalgeometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were...
In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia...
appearance in tropicalgeometry. The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves. The...
have applications to polyhedral combinatorics, linear programming, tropicalgeometry and other areas of mathematics. Given a convex polytope P in Rn, the...
{\displaystyle {\mathcal {G}}.} F1‑geometry has been linked to tropicalgeometry, via the fact that semirings (in particular, tropical semirings) arise as quotients...
more exotic structure to which linear algebra can be applied, see Tropicalgeometry. The system of one equation in one unknown 2 x = 4 {\displaystyle...
simplicial complexes and arise in various areas of polyhedral geometry, such as tropicalgeometry, splines and hyperplane arrangements. A polyhedral complex...
In tropicalgeometry, a tropical projective space is the tropical analog of the classic projective space. Given a module M over the tropical semiring...
reasons, in tropicalgeometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication"...
commutative algebra and algebraic geometry, with an emphasis on toric varieties, Hilbert schemes, and tropicalgeometry. As a student at Burnside High School...
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations...
algebraic geometry, most notably real algebraic geometry, tropicalgeometry and knot theory. Viro developed a "patchworking" technique in algebraic geometry, which...
research involves combinatorial commutative algebra, graph theory, and tropicalgeometry. Chan was inspired to become a violinist as a pre-schooler, seeing...
Her research includes work in algebraic geometry, arithmetic geometry, tropicalgeometry and enumerative geometry. Caporaso earned a laurea from Sapienza...
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory...
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