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Tropical Curves

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Tropical mathematics has two operations, addition, , and multiplication, , which differ from traditional addition and subtraction. Tropical addition is defined as the minimum of the numbers while tropical multiplication is defined as the ordinary sum. For example, and . A power is repeated multiplication; for example, .

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The graph (not shown here) of the tropical polynomial of one variable is simply the lower envelope of the lines , , , and .

"A tropical polynomial function is given as the minimum of a finite set of linear functions. We define the hypersurface to be the set of all points at which this minimum is attained at least twice," in [1].

For example, the surface is represented by the points where occurs in at least two of the terms.

This Demonstration plots the surface of degree 6,

,

which is equivalent to plotting the points where at least two of the terms

give the minimum.

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Contributed by: Whitney Taylor (November 2011)
Open content licensed under CC BY-NC-SA

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Reference

[1] D. Speyer and B. Sturmfels, "Tropical Mathematics," Mathematics Magazine, 82(3), 2009 pp. 163–173.

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