On the number of invariant lines for polynomial systems
Authors:
Zhang Xiang and Ye Yanqian
Journal:
Proc. Amer. Math. Soc. 126 (1998), 22492265
MSC (1991):
Primary 34C05
MathSciNet review:
1476400
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209212) for the proof of the conjecture on the maximum number of invariant straight lines of cubic and quartic polynomial differential systems; and also prove the conjecture in a previous paper of the second author (Qualitative theory of polynomial differential systems, Shanghai ScienceTechnical Publishers, Shanghai, 1995, p. 474) for a certain special case of the degree polynomial systems. Furthermore, we will prove that cubic and quartic differential systems have invariant straight lines along at most six and nine different directions, respectively, and also show that the maximum number of the directions can be obtained.
 1.
Ye Yanqian, Qualitative Theory of Polynomial Differential Systems, Shanghai ScienceTechnical Publisher, Shanghai, 1995.
 2.
Xi
Kang Zhang, The number of integral lines of polynomial systems of
degree three and four, Proceedings of the Conference on Qualitative
Theory of ODE (Chinese) (Nanjing, 1993), 1993, pp. 209–212
(Chinese, with English summary). MR 1259799
(95a:34047)
 3.
Yan
Qian Ye, Sui
Lin Cai, Lan
Sun Chen, Ke
Cheng Huang, Ding
Jun Luo, Zhi
En Ma, Er
Nian Wang, Ming
Shu Wang, and Xin
An Yang, Theory of limit cycles, 2nd ed., Translations of
Mathematical Monographs, vol. 66, American Mathematical Society,
Providence, RI, 1986. Translated from the Chinese by Chi Y. Lo. MR 854278
(88e:58080)
 4.
R.E.Kooij, Some new properties of cubic systems, Delft Univ. Tech., Research Report, preprint, 1991.
 5.
R.E.Kooij, Real cubic systems with four line invariants, Delft Univ. Tech., Research Report, 91 79, 1991.
 6.
G.Darboux, Mémoire sur les equations differentielles algebriques du premier order et du premier degr, Bull. des Sc. Math., 1878, pp.6096; 123144; 151200.
 7.
J.P.Jouanolou, Equations de phaff algebriques, Lect. Notes in Math., 708(1979).
 8.
Suo Guangjian and Sun Jifang, The nth degree differential system with straight line solutions has no limit cycles, Proc. of the Conf. on Ordinary Differential Equations and Contral Theory, Wuhan, 1987.
 9.
Jacek
Sokulski, On the number of invariant lines for polynomial vector
fields, Nonlinearity 9 (1996), no. 2,
479–485. MR 1384487
(96m:34081), http://dx.doi.org/10.1088/09517715/9/2/011
 10.
Dai Guoren, Two estimations of the number of invariant straight lines for nth polynomial differential systems, Acta Mathematica Scientia, 16(1996), 2, 232240.
 11.
J. C. Artes and J. Llibre, On the number of slopes of invariant straight lines for polynomial differential systems, J. Nanjing Univ., Math. Biquarterly, 13(1996), 2, 143149. CMP 97:10
 1.
 Ye Yanqian, Qualitative Theory of Polynomial Differential Systems, Shanghai ScienceTechnical Publisher, Shanghai, 1995.
 2.
 Zhang Xikang, Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, 209212. MR 95a:34047
 3.
 Ye Yanqian et al., Theory of Limit Cycles, Trans.Math.Monographs, Amer. Math. Soc., 66(1986). MR 88e:58080
 4.
 R.E.Kooij, Some new properties of cubic systems, Delft Univ. Tech., Research Report, preprint, 1991.
 5.
 R.E.Kooij, Real cubic systems with four line invariants, Delft Univ. Tech., Research Report, 91 79, 1991.
 6.
 G.Darboux, Mémoire sur les equations differentielles algebriques du premier order et du premier degr, Bull. des Sc. Math., 1878, pp.6096; 123144; 151200.
 7.
 J.P.Jouanolou, Equations de phaff algebriques, Lect. Notes in Math., 708(1979).
 8.
 Suo Guangjian and Sun Jifang, The nth degree differential system with straight line solutions has no limit cycles, Proc. of the Conf. on Ordinary Differential Equations and Contral Theory, Wuhan, 1987.
 9.
 J.Sokulski, On the number of invariant lines of polynomial vector fields, Nonlinearity, 9(1996), 479485. MR 96m:34081
 10.
 Dai Guoren, Two estimations of the number of invariant straight lines for nth polynomial differential systems, Acta Mathematica Scientia, 16(1996), 2, 232240.
 11.
 J. C. Artes and J. Llibre, On the number of slopes of invariant straight lines for polynomial differential systems, J. Nanjing Univ., Math. Biquarterly, 13(1996), 2, 143149. CMP 97:10
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
34C05
Retrieve articles in all journals
with MSC (1991):
34C05
Additional Information
Zhang Xiang
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, China 210097
Email:
xzhang@pine.njnu.edu.cn
Ye Yanqian
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, China 210008
DOI:
http://dx.doi.org/10.1090/S0002993998047108
PII:
S 00029939(98)047108
Keywords:
Polynomial differential system,
invariant line
Received by editor(s):
October 30, 1996
Additional Notes:
The authors were supported by the National Natural Foundation of the People’s Republic of China
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1998
American Mathematical Society
