When do two plane cubic curves have 9 real intersection?

Question

What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an example with 9 real intersections but a characterization.

Answer 1

This "answer" cannot be posted as a comment because it's a bit detailed (though it does not fully answer my question). I just want to say that one of the curves being an M-curve (two connected component in $\mathbb P^2(\mathbb R)$) is not an necessary condiiton!

This is what I did:

• I took 8 random real points in the affine plane namely:

  [0,1],[1,2],[-1,1],[-3,-1],[5,-1],[0,0],[5,0],[-7,3]

• I solved for polynomials in two variables and total degree 3 that define a curve that passes through these points: $$x^3(-233/5250a-33/350b)+x^2y(313/1050a+23/70b)+x^2(1331/5250a+181/350b)+xy^2(807/875a+246/175b)+xy(-883/5250a-83/350b)+x(-83/525a-8/35b)+y^3(-a-b)+y^2a+yb$$ where $a,b$ are any number.

• I tweaked $a,b$ so that I get two curves that are not M-curves. Here is an example the first curve is when $a=1,b=-10$ and the other curve is when $a=10,b=-10$. The part of the curves on top intersect 4 times and the remaining parts intersect 5 times. I hope this is clear.. Nevertheless, it does not answer my question. It just tells me that it is not necessary to be an M-curve to get maximum intersections.