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, .


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.


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




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

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