Real singular points and irreducibility

Question

Let $f\in \mathbb R[X,Y]$ be a polynomial with real coefficients of degree $d$. Now suppose I know that there are at least $(d-1)(d-2)/2+1$ isolated singular points. Does this imply that $f$ is reducible in $\mathbb R[X,Y]$? I know that the condition in general implies that it is reducible over $\mathbb C$ but does this also apply over $\mathbb R$?

Answer 1

No. The idea is to take an irreducible, non-real, polynomial $g \in \mathbb{C}[X,Y]$ and consider the polynomial $f=g \cdot \overline{g}$ which is irreducible over the reals.

For example let $d=2$ and let $f=x^2+y^2$. There is one isolated singularity at the origin, but $f$ is irreducible over the reals.