\(\text{Definition}\)
Tropicalization
"\(a+b\)"\(:=max(x,y)\)
"\(a\)\(\times\)\(b\)"\(:=x+y\)
- Idempotence: "\(x+x\)"\(=x\)
- Convex piecewise linearity: The image of the function consists of a series of linear segments connected at turning points.
- Set of zero points: The set of zero points \(V(f)\) of a tropical polynomial \(f\) is defined as the set of the common maximum points of at least two terms, that is, the set of the "points" at the end of the "line" in the image.
Tropical linuar function
\(g=max(a_0,a_1+x)\)
Then we call \(x=a_0-a_1\) is a root of this function. As you can see, this root is an inflection point of the function \(g\).
Tropical quadratiz function
\(g=max(a_0,a_1x,a_2+2x)\)
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Case 1:
Similar to tropical linuar function, it has \(2\) roots: \(x_1=a_0-a_1,x_2=a_1-a_2\). -
Case2:
In this case, the line \(y=a_1+x\) is always under \(y=a_0\) and \(y=a_2+2x\) ,which means it doesn't show up on the image.
Observe that there exists only one inflection point that \(x=\frac{a_0+a_2}{2}\) .
Multiplicities
Select a portion of the tropical function, multiplicity is defined as \(m-k\) where \(m>k\).
Tropical plane curves
\(f(x,y)=max(a_{i,j}+ix+jy)\) where \(i,j\in Z,a_{i,j}\)\(\in\)\(R\)
A tropical curve defined by \(f\) is the set of points in \(ℝ^2\) where there is maxium is attained at least twice.
Here are some examples.
标签：function,set,draft1,defined,max,tropical,points From： https://www.cnblogs.com/CM-0728/p/18312910