X. Zhang et Yq. Ye, ON THE NUMBER OF INVARIANT LINES FOR POLYNOMIAL SYSTEMS, Proceedings of the American Mathematical Society, 126(8), 1998, pp. 2249-2265
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.