ON THE NUMBER OF INVARIANT LINES FOR POLYNOMIAL SYSTEMS

In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree thre e and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209-212) for the proof of the conjecture on the maximum number of inva riant straight lines of cubic and quartic polynomial differential syst ems; and also prove the conjecture in a previous paper of the second a uthor (Qualitative theory of polynomial differential systems, Shanghai Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain s pecial case of the n degree polynomial systems. Furthermore, we will p rove that cubic and quartic differential systems have invariant straig ht lines along at most six and nine different directions, respectively , and also show that the maximum number of the directions can be obtai ned.