Bivariate Form Sum of Squares

Question

Let $F \in \mathbb{R}[x,y]$ be nonnegative and homogeneous of degree $2n$. Then it can be written as a sum of two squares.

Answer 1

As long as the $x^{2n}$ coefficient is non zero we can write $$f(x,y)=c\prod_{j=1}^n(x-\alpha_j y)(x-\overline{\alpha_j} y)$$ for some complex numbers $\alpha_j$ and $c>0$. Now, write $$\prod_{j=1}^n(x-\alpha_j y)=u(x,y)+iv(x,y)$$ where $u$ and $v$ are real polynomials,....