Polynomial solutions of the Monge-Ampère equation

Autor:	Yu. A. Aminov
Rok vydání:	2014
Předmět:	Combinatorics Minimal polynomial (field theory) Algebra and Number Theory Alternating polynomial Stable polynomial Homogeneous polynomial Polynomial remainder theorem Degree of a polynomial Square-free polynomial Matrix polynomial Mathematics
Zdroj:	Sbornik: Mathematics. 205:1529-1563
ISSN:	1468-4802 1064-5616
DOI:	10.1070/sm2014v205n11abeh004428
Popis:	The question of the existence of polynomial solutions to the Monge-Ampere 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.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::9e6d27783664e3e9fe3b5c3271b33c8f https://doi.org/10.1070/sm2014v205n11abeh004428 Zobrazit plný text záznamu