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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?

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I think it will contradict with the definition of a tropical root of a tropical polynomial with multiplicity k, because it is defined as: Definition 2.4.4. Given a polynomial p(x) with coefficients in A and a root c of the polynomial, the order of c is the number k if p(x) is a multiple of the polynomial (x − c) k but not of the polynomial (x − c) k+1 . But if you are interested in real powers, you can resort to tropical signomial maps. For more info, refer to: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf

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