Open AccessPoincaré index of plane polynomial fields of third and fourth degree
Author(s) -
П. П. Забрейко,
A. V. Krivko-Krasko
Publication year - 2019
Publication title -
vescì nacyânalʹnaj akadèmìì navuk belarusì. seryâ fìzìka-matèmatyčnyh navuk
Language(s) - English
Resource type - Journals
eISSN - 2524-2415
pISSN - 1561-2430
DOI - 10.29235/1561-2430-2019-55-1-22-31
Subject(s) - degree (music) , polynomial , mathematics , plane (geometry) , divisor (algebraic geometry) , zero (linguistics) , degree of a polynomial , singular point of a curve , mathematical analysis , complex plane , reciprocal polynomial , field (mathematics) , matrix polynomial , pure mathematics , geometry , physics , linguistics , philosophy , acoustics
The conditions of isolation of a zero singular point of plane polynomial fields of third and fourth degree are considered in terms of the coefficients of the components of these fields. The isolation conditions depend on the greatest common divisor of the components of polynomial fields: in some cases only on its degree, and in some cases, additionally, on the presence of nonzero real zeros. The reasoning, which allows one to write out the isolation conditions, is based on the concept of the resultant and subresultants of components of plane polynomial fields. If the zero singular point is isolated, its index is calculated through the values of subresultants and coefficients of components.