Geodesic
On Surjective Quadratic Mappings
Matematičeskie zametki, Tome 99 (2016) no. 2, pp. 181-185.

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In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from $\mathbb{R}^n$ to $\mathbb{R}^n$ is studied. For $n=3$, the absence of these zeros is proved.
Keywords: quadratic mapping, quadratic surjective mapping, stable surjectivity. 
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A. V. Arutyunov; S. E. Zhukovskii. On Surjective Quadratic Mappings. Matematičeskie zametki, Tome 99 (2016) no. 2, pp. 181-185. http://geodesic.mathdoc.fr/item/MZM_2016_99_2_a2/

[1] L. L. Dines, “On the mapping of quadratic forms”, Bull. Amer. Math. Soc., 47 (1941) | DOI | MR | Zbl

[2] A. S. Matveev, “O vypuklosti obrazov kvadratichnykh otobrazhenii”, Algebra i analiz, 10:2 (1998), 159–196 | MR | Zbl

[3] B. T. Polyak, “Convexity of quadratic transformations and its use in control and optimization”, J. Optim. Theory Appl., 99:3 (1998), 553–583 | DOI | MR | Zbl

[4] A. A. Agrachev, “Topologiya kvadratichnykh otobrazhenii i gessiany gladkikh otobrazhenii”, Itogi nauki i tekhn. Ser. Algebra. Topol. Geom., 26, VINITI, M., 1988, 85–124 | MR | Zbl

[5] A. V. Arutyunov, “Teorema o neyavnoi funktsii kak realizatsiya printsipa Lagranzha. Anormalnye tochki”, Matem. sb., 191:1 (2000), 3–26 | DOI | MR | Zbl

[6] A. V. Arutyunov, “Gladkie anormalnye zadachi teorii ekstremuma i analiza”, UMN, 67:3 (2012), 3–62 | DOI | MR | Zbl

[7] L. L. Dines, “On the mapping of $n$ quadratic forms”, Bull. Amer. Math. Soc., 48 (1942), 467–471 | DOI | MR | Zbl

[8] L. L. Dines, “On linear combinations of quadratic forms”, Bull. Amer. Math. Soc., 49 (1943), 388–498 | DOI | MR | Zbl

[9] A. A. Agrachev, A. V. Sarychev, “Abnormal sub-Riemannian geodesics: Morse index and rigidity”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13:6 (1996), 635–690 | MR | Zbl

[10] A. V. Arutyunov, “Nekotorye svoistva kvadratichnykh otobrazhenii”, Vestn. Mosk. un-ta. Cer. 15. Vychislit. matem., kibernet., 1999, no. 2, 30–32 | MR | Zbl

[11] A. V. Arutyunov, D. Yu. Karamzin, “Regulyarnye nuli kvadratichnykh otobrazhenii i ikh prilozhenie”, Matem. sb., 202:6 (2011), 3–28 | DOI | MR | Zbl

[12] A. V. Arutyunov, “Neotritsatelnost kvadratichnykh form na peresechenii kvadrik i kvadratichnye otobrazheniya”, Matem. zametki, 84:2 (2008), 163–174 | DOI | MR | Zbl

[13] A. V. Arutyunov, S. E. Zhukovskii, “Suschestvovanie obratnykh otobrazhenii i ikh svoistva”, Differentsialnye uravneniya i topologiya. II, Tr. MIAN, 271, MAIK, M., 2010, 18–28 | MR | Zbl