Proceedings of the American Mathematical Society

Author(s): Zhang Xiang; Ye Yanqian
Journal: Proc. Amer. Math. Soc. 126 (1998), 2249-2265.
MSC (1991): Primary 34C05
Retrieve article in: PDF
This article is available free of charge

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. 209-212) 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 Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain special case of the $n$

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 Science-Technical 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, 209-212. 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 $\sim$ 79, 1991.
6.: G.Darboux, Mémoire sur les equations differentielles algebriques du premier order et du premier degr $\acute e$ , Bull. des Sc. Math., 1878, pp.60-96; 123-144; 151-200.
7.: J.P.Jouanolou, Equations de phaff algebriques, Lect. Notes in Math., 708(1979).
8.: Suo Guangjian and Sun Jifang, The n-th degree differential system with $\frac{(n-1)(n+2)}{2}$ 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), 479-485. MR 96m:34081
10.: Dai Guoren, Two estimations of the number of invariant straight lines for n-th polynomial differential systems, Acta Mathematica Scientia, 16(1996), 2, 232-240.
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, 143-149. CMP 97:10

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34C05

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: 10.1090/S0002-9939-98-04710-8
PII: S 0002-9939(98)04710-8
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
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS