Hilbert's theorem on positive ternary quartics (Q2704210)

In his great paper ``Über die Darstellung definiter Formen als Summe von Formenquadraten'' [Math. Ann. 32, 342-350 (1888; JFM 20.0198.02)] \textit{D. Hilbert} proved that a positive semidefinite ternary quartic (i.e., a homogeneous polynomial with 3 variables and degree 4) with coefficients in the real numbers is a sum of three squares of quadratic forms. The present paper gives a modern proof following the lines of Hilbert's arguments. The basic idea can be outlined easily: Ternary quartics are linear combinations of 15 monomials, hence the set of ternary quartics is identified with the projective spaces \(P^{14}(\mathbb{R})\). The positive definite ternary quartics form a subset, say \(U\), of this projective space. Similarly, a quardratic form is a linear combination of 6 monomials so that the set of triples \([f,g,h]\) of quadratic forms can be identified with the projective space \(P^{17}(\mathbb{R})\). The image of the map NEWLINE\[NEWLINEP^{17}(\mathbb{R})\to P^{14}(\mathbb{R}): [f,g,h]\to f^2+ g^2+ h^2NEWLINE\]NEWLINE is clearly contained in \(U\). The proof shows that the image is actually equal to \(U\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].