Your example of $f(x,y)=\left(y-T_5(x)\right)\left(x-T_5(y)\right)$ is $$ 256\,{x}^{5}{y}^{5}-320\,{x}^{5}{y}^{3}-320\,{x}^{3}{y}^{5}-16\,{x}^{6 }+80\,{x}^{5}y+400\,{x}^{3}{y}^{3}+80\, x{y}^{5}\\-16\,{y}^{6}+20\,{x}^{4 }-100\,{x}^{3}y-100\,x{y}^{3}+20\,{y}^{4}-5\,{x}^{2}+26\,xy-5\,{y}^{2} $$

If I count correctly, it has $28$ regions would that qualify as maximum degree $10?$ If so, then as Pietro points out, $g(x,y)=\prod_{i=1}^{10}\left(a_ix+b_iy+c_i\right)$ will have $56$ regions (if $a_i,b_i,c_i$ are such that the $10$ lines are in general position: no two parallel and no three meeting at a common point). Similar things (as he says) can be done with hyperplane arrangements in higher dimension. You can color the regions $g /gt 0$ white and $g \lt 0$ black so that each region is bounded by regions of the opposite color.

If you want the curve itself to have many disjoint connected components then $g(x,y)+\epsilon$ and $g(x,y)-\epsilon$ are nice to look at. Then all the regions of one color fuse together but each of the others becomes a nicely bordered region. I think that (in the two variable case with a projective viewpoint, at any rate) these achieve that bound given by Harnack's theorem.

Your example of $f(x,y)=\left(y-T_5(x)\right)\left(x-T_5(y)\right)$ is $$ 256\,{x}^{5}{y}^{5}-320\,{x}^{5}{y}^{3}-320\,{x}^{3}{y}^{5}-16\,{x}^{6 }+80\,{x}^{5}y+400\,{x}^{3}{y}^{3}+80\, x{y}^{5}\\-16\,{y}^{6}+20\,{x}^{4 }-100\,{x}^{3}y-100\,x{y}^{3}+20\,{y}^{4}-5\,{x}^{2}+26\,xy-5\,{y}^{2} $$

If I count correctly, it has $28$ regions. Would that qualify as maximum degree $10?$ If so, then as Pietro points out, $g(x,y)=\prod_{i=1}^{10}\left(a_ix+b_iy+c_i\right)$ will have $56$ regions (if $a_i,b_i,c_i$ are such that the $10$ lines are in general position: no two parallel and no three meeting at a common point). Similar things (as he says) can be done with hyperplane arrangements in higher dimension. You can color the regions $g \gt 0$ white and $g \lt 0$ black so that each region is bounded by regions of the opposite color.

If you want the curve itself to have many disjoint connected components then $g(x,y)+\epsilon$ and $g(x,y)-\epsilon$ are nice to look at. Then all the regions of one color fuse together but each of the others becomes a nicely bordered region. I think that (in the two variable case with a projective viewpoint, at any rate) these achieve that bound given by Harnack's theorem.

Your example of $f(x,y)=\left(y-T_5(x)\right)\left(x-T_5(y)\right)$ is $$ 256\,{x}^{5}{y}^{5}-320\,{x}^{5}{y}^{3}-320\,{x}^{3}{y}^{5}-16\,{x}^{6 }+80\,{x}^{5}y+400\,{x}^{3}{y}^{3}+80\, x{y}^{5}\\-16\,{y}^{6}+20\,{x}^{4 }-100\,{x}^{3}y-100\,x{y}^{3}+20\,{y}^{4}-5\,{x}^{2}+26\,xy-5\,{y}^{2} $$

If I count correctly, it has $28$ regions would that qualify as maximum degree $10?$ If so, then as Pietro points out, $g(x,y)=\prod_{i=1}^{10}\left(a_ix+b_iy+c_i\right)$ will have $56$ regions (if $a_i,b_i,c_i$ are such that the $10$ lines are in general position: no two parallel and no three meeting at a common point). Similar things (as he says)can be done with hyperplane arrangements in higher dimension.

If you want the curve itself to have many disjoint components then $g(x,y)+\epsilon$ and $g(x,y)-\epsilon$ are nice to look at.

Your example of $f(x,y)=\left(y-T_5(x)\right)\left(x-T_5(y)\right)$ is $$ 256\,{x}^{5}{y}^{5}-320\,{x}^{5}{y}^{3}-320\,{x}^{3}{y}^{5}-16\,{x}^{6 }+80\,{x}^{5}y+400\,{x}^{3}{y}^{3}+80\, x{y}^{5}\\-16\,{y}^{6}+20\,{x}^{4 }-100\,{x}^{3}y-100\,x{y}^{3}+20\,{y}^{4}-5\,{x}^{2}+26\,xy-5\,{y}^{2} $$

If I count correctly, it has $28$ regions would that qualify as maximum degree $10?$ If so, then as Pietro points out, $g(x,y)=\prod_{i=1}^{10}\left(a_ix+b_iy+c_i\right)$ will have $56$ regions (if $a_i,b_i,c_i$ are such that the $10$ lines are in general position: no two parallel and no three meeting at a common point). Similar things (as he says) can be done with hyperplane arrangements in higher dimension. You can color the regions $g /gt 0$ white and $g \lt 0$ black so that each region is bounded by regions of the opposite color.

If you want the curve itself to have many disjoint connected components then $g(x,y)+\epsilon$ and $g(x,y)-\epsilon$ are nice to look at. Then all the regions of one color fuse together but each of the others becomes a nicely bordered region. I think that (in the two variable case with a projective viewpoint, at any rate) these achieve that bound given by Harnack's theorem.