On the number of invariant lines

for polynomial systems

Authors:
Zhang Xiang and Ye Yanqian

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2249-2265

MSC (1991):
Primary 34C05

DOI:
https://doi.org/10.1090/S0002-9939-98-04710-8

MathSciNet review:
1476400

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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. 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 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 79, 1991.**6.**G.Darboux, Mémoire sur les equations differentielles algebriques du premier order et du premier degr, 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 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**

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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:
https://doi.org/10.1090/S0002-9939-98-04710-8

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

Article copyright:
© Copyright 1998
American Mathematical Society