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Mat. Sb., 2019, Volume 210, Number 12, Pages 3–30 (Mi msb9168)  

The action of the Monge-Ampère operator on polynomials in the plane and its fixed points of polynomial type

Yu. A. Aminov

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine

Abstract: The action of the Monge-Ampère operator on polynomials of degree four in two variables is investigated. Two necessary conditions for the Monge-Ampère equation to have a solution are established. Sufficient conditions for solvability are indicated, which coincide with necessary conditions in certain cases. Invariant submanifolds of the action of the Monge-Ampère operator are found. Closed invariant chains of polynomials are constructed, and all the fixed points having the form of general polynomials of degree four are found.
Bibliography: 9 titles.

Keywords: cone, conic, necessary condition, solvability of equations, invariant set, fixed point.

DOI: https://doi.org/10.4213/sm9168

Full text: PDF file (642 kB)
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References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2019, 210:12, 1663–1689

Bibliographic databases:

UDC: 514.77+517.95
MSC: 35C11, 35G20
Received: 12.09.2018 and 02.04.2019

Citation: Yu. A. Aminov, “The action of the Monge-Ampère operator on polynomials in the plane and its fixed points of polynomial type”, Mat. Sb., 210:12 (2019), 3–30; Sb. Math., 210:12 (2019), 1663–1689

Citation in format AMSBIB
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