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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 a tropical polynomial is defined as the tropical sum (i.e "min" in the real world) of tropical monomials of the form ``$c\odot x_1^{a_1} \odot x_2^{a_2}..\odot x_n^{a_n}$" is for a tuple of positive integers $a_1,a_2,..,a_n$ and $c$ being any arbitrary real number.

  • Is there a way to decide if a tropical polynomial has a factorization (in the tropical sense) which in the real world would mean a specific form like say, $\sum_i \min \{L_{i1},L_{i2} \}$ where $i$ goes over some finite range and the $L$s are affine functions.

  • Lemma 3.1 (page 8) of this review shows how any tropical polynomial as above is associated to a tropical projective variety. Is there a way by which this tropical projective variety might know (easily?) about the tropical polynomial having a factorization of a pre-specified form? (like one said above)

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  • $\begingroup$ if this tropical variety (in fact, hypersurface) is a union of two tropical varieties then the tropical polynomial is a tropical product of two tropical polynomials $\endgroup$
    – Nikita Kalinin
    Jan 18, 2018 at 14:55
  • $\begingroup$ Yes. But here I want to check if a certain tropical function has a specific tropical factorization or not. Do you know how to do this checking on either the function end or the tropical variety end? (How does one go searching if the hypersurface is or is not a union of the "right" kind of tropical varieties which would correspond to this factorization?) $\endgroup$
    – gradstudent
    Jan 22, 2018 at 15:41
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    $\begingroup$ I think 1) compute the natural cellular decomposition of the tropical variety 2) it is easy then to localize "possible" intersections (local intersections like max(x,y,0,x+y)) and then consider connected components of the complement to the possible intersections and try to compose balanced objects using them. $\endgroup$
    – Nikita Kalinin
    Feb 4, 2018 at 16:56
  • $\begingroup$ Okay :) So thats a lot of things that I don't know! Can you give references to read up on each of them, "natural cellular decomposition of a tropical variety", "localize intersections" and "balanced objects". None of this I know! Please give a study route to understand them! $\endgroup$
    – gradstudent
    Feb 5, 2018 at 17:49
  • $\begingroup$ it is faster to explain. 1) cellular decomposition: for each point $x$ of tropical intersection there is a set $S$ of tropical monomials which are maximal at $x$. If you fix $S$ and consider all such points where the set of maximal monomials is $S$, this is a cell. $\endgroup$
    – Nikita Kalinin
    Feb 6, 2018 at 16:07

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