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[WSS22] Extended Newton polytopes of tropical polynomials

Posted 4 months ago
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POSTED BY: Afreen Naz
2 Replies

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POSTED BY: Moderation Team

It's two-dimensional but we can make more plots, we can visualize a two-dimensional tropical curve in a ContourPlot of the tropicalized polynomial.

Manipulate[
 Show[
  ListPlot3D[
   coord3D[Round[c] 2 + 8 x + 5 y, {x, y}],
   Mesh -> None,
   Axes -> True,
   PlotStyle -> {Yellow},
   Filling -> Bottom,
   BoundaryStyle -> Thick,
   PlotLabels -> "Newton Polytope"],
  ListPointPlot3D[
   coord3D[Round[c] 2 + 8 x + 5 y, {x, y}]
   ]], {c, 1, 10}]

This precious article on the Newton Polytopes & those 3D coordinates, the co-efficients, the valuations, and most of all, an in-depth look at Tropical MinPlus algebra. Here, we have used the same polynomial p as above.

Contour Plot

Maybe some other kind of operations... like the one with regard to grouping, distribution, and order of operands. Or to the highest prime of numbers like 3 or 5, also, with regard to the CoefficientRules is this good? Hard-code variables like x and y?

coord3D[poly_, variables_] := 
 Flatten /@ 
  Transpose[{First /@ CoefficientRules[poly, {x, y}], 
    IntegerExponent[#, 2] & /@ 
     Last /@ CoefficientRules[poly, {x, y}]}]
POSTED BY: Dean Gladish
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