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InSimilar questions were studied in the paper,

A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611 .

a similar questions were studied. In particular, for n=3$n=3$ there are necessary and sufficient conditions for a quadratic mapping to be surjective. The paper is written in Russian. Probably, it will be translated this year. If it is interesting, I can write a short summary.

In the paper,

A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611

a similar questions were studied. In particular, for n=3 there are necessary and sufficient conditions for a quadratic mapping to be surjective. The paper is written in Russian. Probably, it will be translated this year. If it is interesting, I can write a short summary.

Similar questions were studied in the paper

A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611 .

In particular, for $n=3$ there are necessary and sufficient conditions for a quadratic mapping to be surjective. The paper is written in Russian. Probably it will be translated this year. If it is interesting, I can write a short summary.

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In the paper,

A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611

a similar questions were studied. In particular, for n=3 there are necessary and sufficient conditions for a quadratic mapping to be surjective. The paper is written in Russian. Probably, it will be translated this year. If it is interesting, I can write a short summary.