Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$.
Page 281 says: "Let us denote $\nabla(n, d, \mathbb{C}) \subset \mathbb{C}(n,d)$ the set of singular polynomials of degree $d$ in $n$ variables, over the complex numbers. That is, $\nabla(n, d, \mathbb{C}) = \{F \in \mathbb{C}(n, d) | \text{there exists } x \in \mathbb{C}^n-\{0\}, \frac{\partial F}{\partial x_i}(x)=0, \forall i\}$. It is known that $\nabla(n, d, \mathbb{C})$ is an irreducible algebraic hypersurface of degree $D = n(d − 1)^{n−1}$ defined over the rational numbers. Therefore, there exists a polynomial (unique up to multiplicative constant) $\Delta = \Delta(n, d)$ called the discriminant, such that $\nabla(n, d, \mathbb{C}) = \{F \in \mathbb{C}(n, d) | \Delta(F) = 0\}$".
For simplicity, let us concentrate on the case $n=2$.
(1) Could one please explain or give a reference to "Therefore"?
(2) More important (to me, at the moment), given a homogeneous polynomial of degree $d$ in two variables $x,y$ over $\mathbb{R}$, how to find $\Delta$? In this case, its degree $D$ should be $2(d-1)$.
Thank you very much!