# How to plot a 3D-subspace of a $\mathbb{C}^2$ curve?

I am trying to visualize the intersection of a complex curve in $$\mathbb{C}^2$$ with various 3D subspaces but I am not sure how to get Mathematica to do it (so far I have been able to visualize intersections with 2D subspaces using ReIm and ContourPlot). For example, say I have the curve $$p^4+qp^2+1=q^8$$ in $$\mathbb{C^2}$$, how would I get Mathematica to plot all values where $$p \in \mathbb{R}$$ and $$q \in \mathbb{C}$$?

• Can you show an example anywhere from the web or a paper? Commented Sep 8, 2023 at 21:02
• People here generally like users to post code as copyable Mathematica code as well as images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find the meta Q&A, How to copy code from Mathematica so it looks good on this site, helpful Commented Sep 8, 2023 at 21:25

Using the method in Plotting implicitly-defined space curves

Block[{F},
F[p_, q_] := p^4 + q*p^2 - q^8 + 1;
ContourPlot3D[
ComplexExpand@ReIm@F[pR + I*pI, qR + I*qI] /. pI -> 0 // Evaluate,
{pR, -4, 4}, {qR, -4, 4}, {qI, -2, 2},
MaxRecursion -> 3, PlotPoints -> 25,
Contours -> {0}, ContourStyle -> Opacity[0], Mesh -> None,
BoundaryStyle -> {1 -> None,
2 -> None, {1, 2} -> {{Green, Tube[.03]}}}, Boxed -> False]
]