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.

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}]}]

Analyze polynomials? Using the MinPlus algebra which removes and mainly focuses on Newton Polytopes in the context of Tropical Geometry, don't say that combines algebraic geometry and combinatorics. Look at tropicalization, also known as the degeneration of algebraic varieties into tropical varieties. A multivariate polynomial is associated with integral polytopes and integral polytopes are associated with and can be used to study the polynomial's behavior. Remove duplicates, and explain the concept of MinPlus algebra and Tropical Semiring; a polynomial can be converted into a tropical polynomial consolidating this computationally universal aspect and approach of changing the operations of multiplication to addition and addition to minimum, moving on it's unprecedented. It's like the jacuzzi of vitality.

polyList = {
   {x^2 + y^2, {x, y}},
   {5 x^3 + 3 y^2 + 7 x y, {x, y}},
   {3 x^2 + 2 y z + z^2, {x, y, z}},
   {2 x^2 + 4 y^3 + x*y + 6, {x, y}},
   {4 x^2 + 6 x y + 7 y^2 + 5, {x, y}},
   {16 x^3 + 8 y^2 + 4 x*y + 2, {x, y}},
   {x^3 + y^3 + z^3 + 3 x y z, {x, y, z}},
   {x^4 + y^4 + x^2*y^2 + x + y + 1, {x, y}},
   {x^3 + z^3 + 2 x y z^2 + x y z + 1, {x, y, z}},
   {16 x^2 + 6 x y + 7 y^2 + 7 x + 5 y + 2, {x, y}},
   {x^3 + 2*x^2*y + 3*y^3 + x + y + 1, {x, y}},
   {4 x^3 + 2 y^2 + z^2 + 2 x*y*z + 1, {x, y, z}},
   {4*x^2 + 2*x*y + y^2 + 3*x + 2*y + 1, {x, y}},
   {x^4 + y^4 + z^4 + 4 x y z w + w^4, {x, y, z, w}},
   {3*x^2 + 5*x*y^2 + 7*y^3 + 2*x + y + 1, {x, y}},
   {2*x^2 + 3*y^2 + 4*z^2 + 5*x*y*z + 6, {x, y, z}},
   {8 x^3 + 6 x^2 y + 7 y^3 + 7 x^2 + 5 y^2 + 2 x + 2 y, {x, y}},
   {2*x^3 + 3*x^2*y + 4*x*y^2 + y^3 + x^2 + x*y + y^2 + x + y + 1, {x,
      y}},
   {x^3 + y^3 + z^3 + 3*x^2*y + 2*x*y*z + y^2*z + x + y + z + 1, {x, 
     y, z}},
   {8 x^3 + 4 x^2 y + 6 y^2 z + 2 z^3 + 5 x^2 + 7 y^2 + 3 z^2 + x + 
     y + z, {x, y, z}},
   {10 x^3 + 5 x^2 y + 7 y^2 z + 3 z^3 + 8 x^2 + 4 y^2 + 2 z^2 + 
     5 x + 7 y + 3 z, {x, y, z}},
   {7*x^3 + 3*x*y + y^2 + 6*y + 9, {x, y}}, {3 x^2 + 6 y^2 + 2 z^2 + 
     7 x + 3 y + 4, {x, y, z}},
   {10 x^4 + 5 x^3 y + 7 x^2 y^2 + 3 x y^3 + y^4 + 8 x^3 + 4 x^2 y + 
     2 x y^2 + y^3 + 5 x^2 + 7 x y + 3 y^2 + x + y, {x, y}}};
NewtonPolytope[poly_, variables_, style_] := 
 Module[{tpoly, p, cs, P2 = poly}, 
  p = First /@ CoefficientRules[P2, variables];
  cs = IntegerExponent[#, 2] & /@ 
    Last /@ CoefficientRules[P2, variables];
  tpoly = Flatten /@ Transpose[{p, cs}];
  Show[ListPlot3D[tpoly, style, Mesh -> All, Axes -> True, 
    Filling -> Bottom], ListPointPlot3D[tpoly]]]
commonStyle = {PlotStyle -> {Thickness[0.004], Blue}, 
   ColorFunction -> Function[{x, y, z}, Hue[1.0 z]], MeshStyle -> Red,
    ImageSize -> Large};
plots = Table[
   NewtonPolytope[First[i], Last[i], commonStyle], {i, polyList}];
GraphicsGrid[Partition[plots, 5], Frame -> None]

The transformation, give me a tutorial who does a deep dive into the topic of the 2-adic valuation, the technique used to find the right coefficients of a Tropical Polynomial like it's going out of style, precisely. For the 2-adic valuation for a number n that wouldn't be the same without being defined as the highest power of a prime p that divides n, no need to just provide some examples. Give me some code snippets to provide some grammatical interpretability and illustrate this universe concept. The splendid Newton Polytope of a given polynomial involves the people who labor and come out of the Earth just to get go form 3D coordinates, get exponent vectors and extract values from the valuation to form them in the first place, which springs forth the visualization of the Newton Polytope's opulence, in a three-dimensional plot.

poly = x^2 + y^2 - z^2;
MinPlusPoly[poly_] := 
  ReplaceAll[poly, {Plus -> Min, Times -> Plus, Power -> Times}];
ContourPlot3D[MinPlusPoly[poly], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
 Mesh -> None]

What would happen if I tried to provide a substitution, or an in-depth exploration of Tropical Geometry and Newton Polytopes? That seems to target the audience who resides within the edifice of computational geometry and mathematical programming. @Afreen Naz reading is a joy, full of examples of mathematical software for geometric visualization and algebraic computation.

MinPlus[poly_] := 
  ReplaceAll[poly, {Plus -> Min, Times -> Plus, Power -> Times}];
polyList = {x^2 + y^2, 5 x^3 + 3 y^2 + 7 x y, x^2 + 2*y + 3, 
   8 x^3 + 2 y^2 + 5, 8 x + 2 y + 5, 
   16 x^2 + 6 x y + 7 y^2 + 7 x + 5 y + 2, 
   8 x^2 + 2 x y + 3 y^2 + 4 x + 5 y + 6, 7 x^2 + 6 y^2 + 3 x y};
plots2D = 
  Table[ContourPlot[MinPlus[poly], {x, -10, 10}, {y, -10, 10}, 
    ExclusionsStyle -> Black], {poly, Most[polyList]}];
poly3D = x^2 + y^2 - z^2;
plot3D = 
  ContourPlot3D[MinPlus[poly3D], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   Mesh -> None];
allPlots = Join[plots2D, {plot3D}];
GraphicsGrid[Partition[allPlots, 3]]

This new perspective has been a musical masterpiece in addressing enumerative problems. The application of this opportune discipline to physics & mathematics, hints at its interdisciplinary nature which is characteristically too big for the bottle of the study of particle motion using tropical shapes. It's more sort of incremental growth and there's certainly plenty of things that get done incrementally. And of course when you're dealing with incremental high growth there isn't a well-defined part you go on, you kind of have to invent things for yourself. What questions are worth asking?