Do exponents in tropical polynomials have to be integers?

Question

Recently I've been learning about tropical geometry, and every time I see a definition of a tropical polynomial in e.g. $k$ variables $x_1,x_2,...,x_k$ such as $\bigoplus_{i=1}^n a_i x_1^{b_{i1}}x_2^{b_{i2}}...x_k^{b_{ik}}$, it usually defines the exponents to be integral. Would it be incorrect to define a tropical polynomial with say rational or real valued exponents? Would the object we get by doing so lose any of the nice properties of tropical polynomials? As far as I can tell the definitions would still work out fine.

Answer 1

The "max-plus" operations (replacing addition with maximum and multiplication with addition) really come from a limiting process. For an expression $f(x_1,\dots,x_n)$ in $n$ variables, its tropicalization is the limit:

$$ \lim_{t \to \infty} \frac{1}{t} \, \log f(e^{tx_1},\dots,e^{tx_n}) $$

If you take $f(x,y) = x+y$, then this limit evaluates to $\mathrm{max}(x,y)$, and if $f(x,y) = xy$, then the limit is $x+y$. Both limits are very easy to compute using L'Hospital's rule.

Exponents become multiplication: if you take $f(x) = x^p$, then

$$ \lim_{t \to \infty} \frac{1}{t} \, \log \Big( (e^{tx})^p \Big) = px $$

This clearly does not depend on $p$ being an integer, so the "tropicalization" of non-integer exponents makes sense in the same way.