Can we have “tropical polynomials” with arbitrary real powers?

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a tropical monomial $$c\odot x_1^{a_1} \odot x_2^{a_2}..\odot x_n^{a_n}$$" for a tuple of positive integers $$a_1,a_2,..,a_n$$ is derived from the fundamental definition of a tropical product as, $$x\odot y = x+y$$. Hence literally this makes sense only for $$a_i$$s being positive integers. And then the review says that obviously this tropical monomial is the $$\mathbb{R}^n \rightarrow \mathbb{R}$$ affine map, $$c +\sum_{i=1}^n a_ix_i$$.

• Now if I take this affine map meaning as fundamental then can I as well think of $$c\odot x_1^{a_1} \odot x_2^{a_2}..\odot x_n^{a_n}$$" (and hence a generalized tropical polynomial) for $$a_i$$s arbitrary real numbers - since the affine map meaning for them would be well-defined for any of the monomials?

• And if the above is true and feasible then does the subsequent definition of a tropical hypersurface" given on that page also make sense for such generalized notions of a tropical polynomial?

Is there any standard literature which deals with this generalization?