Tropical Math - Arithmetic/Algebra

 

I just heard this term the other day and the first thing my mind did was to picture numbers lying on the beach, sipping drinks, while sitting under umbrellas. Yes, I know it's not that but there are certain visuals we associate with certain words.  

Tropical math actually covers two different mathematical  sub groups - Algebra and Geometry.  It began developing around the beginning of this century and the adjective "tropical" came from several French mathematicians including Jean-Eric Pin.  

It is based on the "tropical semi-ring" which uses a set of real numbers with the additional element of infinity.  The tropical sum is defined as the sum of their minimums while the product is actually their sum. An example of a tropical sum for 4 and 8 is 4 because 4 is the minimum of both numbers while the product is 12 because 4 + 8 = 12. The notation used for tropical sum is a circle with a cross in the middle and tropical product is indicated by a circle with a dot inside it.  

What makes this even more interesting is that these two operations are actually commutative and can have the distributive property applied to them.  Furthermore, infinity is considered a natural element of addition and zero is a natural element for multiplication.  Unfortunately, subtraction within tropical arithmetic is a bit harder because there is no value for the phrase "10 - 3" thus it is best to stick with addition and multiplication.

As far as polynomials go, tropical monomials are actually linear functions with integer coefficients and a tropical polynomial is actually defined as "a finite linear functions with integer coefficients". Tropical polynomials are also continuous, composed of a piecewise of linear functions and it is concave. In addition, the Fundamental Theorem of Algebra apply to tropical linear functions.

Furthermore, curves in tropical algebra is shown in a hypersurface composed of all "roots" of the polynomials.  This can be extended to polynomials in two variables in which the curve is contained in the plane of real numbers squared with both bounded and unbounded edges.  In addition, the slopes are rational and if the sum of all vectors is taken, the result is zero.

When graphing such polynomials, one sees three half-rays heading off in three different directions. The degree of the polynomial determines the number of half-rays, vertices, bounded and unbounded edges but they half-rays tend to head off to the north, east, and southwestern directions.  This leads to the idea of linear spaces in which solving linear equations requires a person to determine the intersections of a certain number of hyperplanes.  

This is a brief look at tropical arithmetic and algebra but there is also the field on tropical geometry.  I'll be providing a short look at tropical geometry on Wednesday.  Let me know what you think, I'd love to hear. If you want to learn more about this check the internet for some really interesting reading.  Let me know what you think, I'd love to hear.  Have a great day.