Polynomial curves $F(x,y)$ giving rise to loops

Question

I know that the cubic curve defined by $F(x,y) = y^2 - x^3 - x^2$ gives rise to a loop:

Is there a way to check if a cubic curve defined by $F(x,y) = y^2 + ay + bx^3 + cx^2 + dx + e$ exhibits a loop? Can there be more than one loop?

What happens when we allow mixed terms, i.e. $F(x,y) = y^2 + ay + bx^3 + cx^2 + dx + e + fxy$ or even $F(x,y) = y^2 + ay + bx^3 + cx^2 + dx + e + fxy + gx^2y + hxy^2$?

If we allow arbitrary polynomials over $x,y$ how does the maximal number of loops depend on the degree(s) of the polynomial?

Finally: How would one classify the curves according to their "loop structure" which considers the number of loops but possibly distinguishes between these curves which all have two loops:

And: How could one systematically construct a polynomial giving rise to a curve with a given loop structure, e.g. one of the three above.

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(Do my tags make sense? Which ones would be better?)

Answer 1

At a crossover point, $dy/dx=F_x/F_y$ is not defined so $F_x=F_y=0$. An example with $mn$ crossing points is $F(x,y)=g(x)h(y)$ where $g(x)$ has $m$ zeros and $h(y)$ has $n$ zeros.
An example for your middle sketch is $$y^2 = -(x+1)^2x^2(x-1)$$ which has crossings at $(-1,0)$ and $(0,0)$.

Answer 2

This is what I accidentally found in Marcel Bergers Geometry Revealed (p. 287). It's about closed curves (while I had in mind open curves):