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I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says


So why are ordinary linear functions a natural analogue for the exponential function in a tropical semiring?

(Note: an intuitive answer is fine. I don't know any tropical geometry and only basic algebraic geometry, so not necessarily looking for a very technical response, though that would be awesome as well.

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