StPlus[a,b]:= Max[a,b];
StTimes[a,b]:= Plus[a,b].
For such new "+" and "*" operators, -Infinity is the addition zero and 0 is the multiplication unit, since StPlus[a, -Infinity] = a and StTimes[a, 0] = a. However, there is no substraction that could be defined on this field (since the new addition is idempotent, StPlus[x,x]=x. Therefore, such a field is called semifield.
More interesting is the polynomials defined on this semifield.
P[x,y]:=StSum[a[[j]][[k]]*(x^j)*(y^k), {j,1,m},{k,1,n}]
= Max[Sequence@@Table[j*x + ky + a[[j]][[k]],{j,1,m},{k,1,n}]]
The tropical curve C defined by P consists of those points (x,y) which belongs to R^2, where P is not differentiable. "In other words, C is the locus where the maximum is assumed by more than one of the monomials of p."
For example, the following graph represents two tropical curves.
The blue curve is defined by the polynomial, StPlus[x, y, 1] while the red curve is given by StPlus[StTimes[2,x], StTimes[3,y], 5]. This seems a very interesting topic. So far I could only understand these. For higher order polynomials, more interesting features appear and more useful in some applications.
