WE CONSIDER POLYNOMIAL GENERALIZED LIÉNARD SYSTEMS THAT COME FROM SECOND ORDER POLYNOMIAL PERTURBATIONS OF A LINEAR CENTER. FOR THESE SYSTEMS WE FOUND A GENERIC (OPEN AND DENSE) SUBSET IN THE SPACE OF PERTURBATIONS OF DEGREE 2L, WITH , SUCH THAT EACH ASSOCIATED PERTURBED GENERALIZED LIÉNARD SYSTEM HAS AT MOST MEDIUM LIMIT CYCLES. MOREOVER, WE SHOW THAT THIS UPPER BOUND IS REACHED. IN ADDITION, SOME CONCRETE EXAMPLES ARE GIVEN.