TY - JOUR
ID - 6266
TI - Some new families of definite polynomials and the composition conjectures
JO - Computational Methods for Differential Equations
JA - CMDE
LA - en
SN - 2345-3982
AU - Shafeii Lashkarian, Razie
AU - Behmardi Sharifabad, Dariush
AD - Department of Basic science, Hashtgerd Branch,
Islamic Azad University, Alborz, Iran
AD - Department of Mathematics,
Alzahra university, Tehran, Iran
Y1 - 2017
PY - 2017
VL - 5
IS - 3
SP - 214
EP - 223
KW - Abel equation
KW - composition condition
KW - composition conjecture
KW - definite polynomial
KW - moment
DO -
N2 - The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjecture and the moment vanishing problem ask for that the composition condition is a necessary condition to have the center or vanishing the moments. It is not known that if there exist examples of polynomials that satisfy the double moment conditions but don't satisfy the composition condition. In this paper we consider some composition conjectures and give some families of definite polynomials for which vanishing of the moments and the composition condition are equivalent. Our methods are based on a decomposition method for continuous functions. We give an orthogonal basis for the family of continuous functions and study the conjecture in terms of this decomposition.
UR - https://cmde.tabrizu.ac.ir/article_6266.html
L1 - https://cmde.tabrizu.ac.ir/article_6266_311deb2c575c0be337945cb0844190c3.pdf
ER -