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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!

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    $\begingroup$ (1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $\Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $\mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $\Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant $\endgroup$ – Tabes Bridges Jul 15 '18 at 21:15
  • $\begingroup$ Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e \in \mathbb{R}$? $\endgroup$ – user237522 Jul 15 '18 at 21:27
  • $\begingroup$ Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-\frac{b}{2}\frac{b}{2} > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $\Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$ $\endgroup$ – user237522 Jul 15 '18 at 22:20

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