How to plot the zero set of certain functions?

Question

I'm working with functions that look like $$P(z,w)=\sum_{|i|+|j|\leq n}c_{ij}\cdot z^i w^j.$$ Where the exponents $i,j\in\mathbb{Z}$ and $c_{ij}\in\mathbb{R}$. I'm interested in plotting the image of the set $$\{(z,w)\in\mathbb{R}^2:P(z,w)=0\}$$ under the map $(z,w)\mapsto (\log{|z|},\log{|w|})$. I tried something and it works but it seems very slow and just not the best way to do it. My try:

Ps[z_, w_] := 5 + 1/z + 1/w + z + w;
r = ImplicitRegion[Ps[z, w] == 0, {z, w}];
rt = TransformedRegion[r, {Log@Abs@Indexed[#, 1], Log@Abs@Indexed[#, 2]}&];
RegionPlot[rt, PlotRange -> {{-3, 3}, {-3, 3}}]

After some effort it results in

which seems to be correct. Is there a better and faster way to make plots like this?

Answer 1

You could just use ContourPlot with the different branches of the inverse functions of your map:

Show@Table[
  ContourPlot[
   Ps[sz Exp[z], sw Exp[w]] == 0, {z, -3, 3}, {w, -3, 3}], 
  {sz, {1, -1}}, {sw, {1, -1}}]