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I'm a deep learning researcher, and these days studying algebraic geometry for my research and for personal interest.

I'm noob to this field, and I found a research area called tropical geometry (TG).

In TG, multiplication and summation is defined as:

$$x\oplus y = \max\{x,y\}$$ $$x\odot y= x + y$$

I don't get the motivation of these operations.

Up to my opinion, these operations makes sense if we define

$$x\oplus y = \lim_{t\to\infty}\log_t(t^x+t^y)=\max\{x,y\}$$

$$x\odot y = \lim_{t\to\infty}\log_t(t^x\cdot t^y)=(x+y)\lim_{t\to\infty}\log_tt=x+y$$

But I can't get why operations should defined like this.

Is there any motivation or motivating example?

Ryoungwoo Jang
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    The justification for defining the operations is indeed the log-exp limit you mention. This is (apparently) useful for studying the large-scale behaviour of algebraic varieties. – Zhen Lin May 15 '22 at 05:12
  • @ZhenLin Thx. My exact question was whether there exist any motivating example. – Ryoungwoo Jang May 15 '22 at 05:17
  • Maybe this link is of interest for you ? or this one : https://www.quora.com/What-is-the-significance-of-tropical-geometry – Jean Marie May 15 '22 at 06:06

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