Notes towards a constructive proof of Hilbert's theorem on ternary quartics (Q2704207)

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. But he gave no method for finding an explicit representation and no estimate for the number of essentially different ways to do it. These are the issues that are addressed in the present paper. NEWLINENEWLINENEWLINEBoth the general situation and some special cases are discussed. Some progress is made in the general case with two different methods, namely the use of Gram matrices and an analysis of quartics that can be written as sums of two squares of quadratic forms. The importance of the latter method is due to the simple observation that a quartic \(p\) has a representation \(f^2+ g^2+ h^2\) with quadratic forms \(f\), \(g\) and \(h\) if and only if \(p-f^2= g^2+h^2\). Finally, a complete answer is given in the case of a quartic \(X^4+ F(Y,Z)\), where \(F\) is a positive semidefinite binary quartic.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].