Let <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> be a foliation in the projective space of dimension two with a first integral of the type <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, where F and G are two polynomials on an affine coordinate, <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img3.gif"> = <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img4.gif"> and g.c.d.(p, q) = 1. Let z be a nondegenerate critical point of <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, which is a center singularity of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">, and <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> be a deformation of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> in the space of foliations of degree deg(<img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">) such that its unique deformed singularity <img src="http:/img/fbpe/aabc/v73n2/zt.gif" alt="zt.gif (118 bytes)"> near z persists in being a center. We will prove that the foliation <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> has a first integral of the same type of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">. Using the arguments of the proof of this result we will give a lower bound for the maximum number of limit cycles of real polynomial differential equations of a fixed degree in the real plane.
Holomorphic foliation; limit cycle; center singularity
