Polynomial solutions of the Monge-Ampère equation
Journal Article
·
· Sbornik. Mathematics
- B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov (Ukraine)
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
- OSTI ID:
- 22364218
- Journal Information:
- Sbornik. Mathematics, Vol. 205, Issue 11; Other Information: Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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