Corrigendum to “On the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree” [Nonlinear Anal. 69 (12) (2008) 4461–4469]

In the paper [B. García, J. Llibre, J.S. Pérez del Río, On the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree, Nonlinear Analysis 69 (12) (2008) 4461–4469] we studied the number of limit cycles that bifurcate from the periodic orbits of the center View the MathML sourceẋ=−yR(x,y), View the MathML sourceẏ=xR(x,y) where RR is a convenient polynomial of degree 2, when we perturb it inside the class of all polynomial differential systems of degree nn. Actually, the number obtained is not correct and we can now prove that its true value is 2[(n−1)/2]+12[(n−1)/2]+1 instead of ([(n−1)/2]+4)([(n−1)/2]+1)/2−1([(n−1)/2]+4)([(n−1)/2]+1)/2−1. In [1] the proof of Theorem 1 is not completely correct, because its proof, as it is stated, requires that the functions Sk(r)r2j−2Sk(r)r2j−2 for j=0,1,…,[(n−1)/2]j=0,1,…,[(n−1)/2] and k=0,1,…,j+1k=0,1,…,j+1 be linearly independent, and for proving this fact it is not sufficient to show that these functions are independent with jj fixed (Proposition 13) and with kk fixed (a consequence of Corollary 11), as we did in the article. Therefore the number of linearly independent functions is incorrect and, consequently, the statement of Theorem 1 must be changed. Thus in Theorem 1 the value of N(n)N(n) must be 2[(n−1)/2]+12[(n−1)/2]+1 instead of ([(n−1)/2]+4)([(n−1)/2]+1)/2−1([(n−1)/2]+4)([(n−1)/2]+1)/2−1. In short, the correct statement of Theorem 1 is the following one.