Last week, professor Bernd Sturmfels from the University of California Berkeley delivered a talk on tropical mathematics, a unique area of study that offers novel approaches to basic problem solving.
Tropical arithmetic, for example, differs from regular arithmetic in that the addition and multiplication functions take on new definitions. The result of the tropical addition of two numbers is the smaller number (i.e. 3 + 10 = 3), and the result of tropical multiplication of two numbers is the arithmetic sum (i.e. 3 * 10 = 13).
These laws apply to polynomials as well. For example, if we consider the polynomial P(x) = ax2 + bx + c using tropical arithmetic, we would rewrite this polynomial as: min(a+x+x, b+x, c).In other words, depending on the values of the coefficients and the value of x, the answer would be the minimum value of (a+x+x), (b+x), and (c).
In accordance with these polynomial rules, the root of a polynomial in tropical mathematics also takes on an interesting definition. In tropical mathematics, the root is defined as the x value where the minimum value occurs twice; this is in clear contrast to the “normal” consideration of a polynomial’s root as the value of x where the polynomial evaluates to zero. For example, consider the polynomial 2x3+3x2+5x+11, which can be rewritten tropically as min(2+x+x+x, 3+x+x, 5+x, 11). In this case, x = 1 would be a root, because subbing in 1 for x would yield min(2+1+1+1, 3+1+1, 5+1, 11) = min(5, 5, 6, 11). Since the minimum value 5 occurs twice, x = 1 is a root.
While tropical mathematics is predominantly theoretical today, it is steadily developing into an area of study with diverse applications. According to Sturmfels, one potentially useful application of tropical mathematics would be in evolutionary biology. When two matrices are multiplied tropically, minimization is clearly visible and easy to achieve; for example, consider a matrix where each element represents the distance between two points. If a single element were examined (i.e. a13 as the distance between given points 1 and 3), squaring the matrix would yield the shortest possible distance between those two points.
The power of tropical matrices becomes even clearer when large quantities of data are manipulated with computer algorithms. Because the multiplication process is both time and memory efficient, evolutionary biologists hope to use tropical mathematics to retrace the likely paths of evolution by determining the shortest evolutionary distance between species.