The Dines theorem and some other properties of quadratic mappings (Q904388)

The author examines real homogeneous quadratic mappings from \(\mathbb{R}^n\) to \(\mathbb{R}^2\). It is known that the image of such a mapping is always convex. A proof of the convexity of the image based on the quadratic extremum principle is given. The following fact is observed: If the quadratic mapping \(Q\) is surjective and \(n>2+ \dim\ker Q\), then there exists a regular zero of \(Q\). A certain criterion of the linear dependence of quadratic forms is also stated.