# How to plot implicit region that contains Solve

Suppose that I have a complicated function F(a,x,y) and I want to plot the 2D region of (x,y) points such that F(a,x,y)=0 has a solution. How would that be possible in mathematica?

I would think that such a plot requires something on the lines of ImplicitRegion[Solve[F[a,x,y]==0,a],{x,y}] (of course, this code won't work).

EDIT: A simple example would be something like F[a,x,y]:=a^2 +x. Then F[a,x,y]==0 would only have a solution if x<=0 (assuming a,x,y are all Reals) and then I would like to plot the region x<=0 (with something like RegionPlot)

• Thank you for adding an example. I'm afraid that I am still confused though. In your case of $F=a^2+x$, then $F=0$ is not satisfied by any value of $x<0$, but instead by exactly one value of $x$, i.e. $x=-a^2$, for each value of $a$. That's not a region, is it? Perhaps we really do need to see the complicated function. Jan 20 '20 at 1:19
• @MarcoB So if $x=-1,y=0$, then Solve[F[a,x,y]==0,a,Reals] returns a solution, namely $a=\pm 1$. Therefore, in my $(x,y)$ plot, the point $(x,y)=(-1,0)$ should be my colored region. If $x=1,y=0$, then Solve[F[a,x,y]==0,a,Reals] does not return a solution and thus in my $(x,y)$ plot, the point $(x,y)=(1,0)$ should NOT be my colored region. Repeat this thought process for all conceivable $(x,y)$ points and then we obtain a region Jan 20 '20 at 1:24

Get a fast impression e.g. for a^2 + x - y^2 == 0 with

ContourPlot3D[a^2 + x - y^2 == 0, {x, -3, 3}, {y, -3, 3}, {a, -4, 4},
ViewPoint -> {0, 0, Infinity}, AxesLabel -> {x, y},
MeshFunctions -> Function[{x, y, a}, a], Mesh -> {Range[-3, 3]}]


reg = Resolve[Exists[a, x^2 + a^2 - y == 0], Reals] //
ImplicitRegion[#, {x, y}] &
(*  ImplicitRegion[x^2 - y <= 0, {x, y}]  *)

RegionPlot[reg]


Of course, transcendental equations are not always possible for solvers to solve. And high-degree equations can be a challenge, too.