Geodesic
The Dines theorem and some other properties of quadratic mappings
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1661-1669

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Real homogeneous quadratic mappings from $\mathbb{R}^n$ to $\mathbb{R}^2$ are examined. 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 noted: If the quadratic mapping $Q$ is surjective and $n>2+\mathrm{dim\,ker\,}Q$, then there exists a regular zero of $Q$. A certain criterion of the linear dependence of quadratic forms is also stated. 
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     author = {D. Yu. Karamzin},
     title = {The {Dines} theorem and some other properties of quadratic mappings},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1661--1669},
     publisher = {mathdoc},
     volume = {55},
     number = {10},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_10_a4/}
}
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D. Yu. Karamzin. The Dines theorem and some other properties of quadratic mappings. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 10, pp. 1661-1669. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_10_a4/