Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials Original Research Article

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré–Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17–30] the Poincaré–Liapunov constants are calculated for polynomial systems of the form View the MathML sourceẋ=−y+Pn(x,y), View the MathML sourceẏ=x+Qn(x,y), where PnPn and QnQn are homogeneous polynomials of degree nn. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré–Liapunov constants, through the study of the highest fine focus order for n=4n=4 and n=5n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré–Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870–1877]. Moreover, the computational problems which appear in the computation of the Poincaré–Liapunov constants and the determination of the number of functionally independent ones are also discussed.