Geodesic
An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 171-179 
V. B. Tlyachev; A. D. Ushkho; D. S. Ushkho. An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 171-179. http://geodesic.mathdoc.fr/item/ISU_2015_15_2_a6/
@article{ISU_2015_15_2_a6,
     author = {V. B. Tlyachev and A. D. Ushkho and D. S. Ushkho},
     title = {An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {171--179},
     year = {2015},
     volume = {15},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2015_15_2_a6/}
}
TY  - JOUR
AU  - V. B. Tlyachev
AU  - A. D. Ushkho
AU  - D. S. Ushkho
TI  - An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2015
SP  - 171
EP  - 179
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ISU_2015_15_2_a6/
LA  - ru
ID  - ISU_2015_15_2_a6
ER  - 
%0 Journal Article
%A V. B. Tlyachev
%A A. D. Ushkho
%A D. S. Ushkho
%T An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2015
%P 171-179
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/ISU_2015_15_2_a6/
%G ru
%F ISU_2015_15_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that the $n$-th degree polynomial vector field in the plane has at most $2n + 1$ ($2n + 2$) invariant straight lines when $n$ is even (odd) and $n\geq 3$ if it has a singular point for which $n + 1$ invariant straight lines and $n$ parallel invariant straight lines with a certain angular coefficient are incident.

[1] Joan C. A., Grunbaum B., Llibre J., “On the number of invariant straight lines for polynomial differential systems”, Pacific J. Math., 184:2 (1998), 207–230 | DOI | MR | Zbl

[2] Druzhkova T. A., Algebraic and differential equations with algebraic integrals, v. 1, Nizhny Novgorod Univ. Press, Nizhny Novgorod, 2005, 37 pp. (in Russian)

[3] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Qualitative Theory of Second-Order Dynamic Systems, John Wiley, Jerusalem–New York, 1973, 524 pp. | MR | MR | Zbl

[4] Dolov M. V., Chistyakova S. A., “On linear partial integrals of polynomial vector fields of the fourth degree with degenerate infinity, I”, Bulletin of the University of Nizhny Novgorod, 2010, no. 6, 132–137 (in Russian)

[5] Dolov M. V., Chistyakova S. A., “On linear partial integrals of polynomial vector fields of the fourth degree with degenerate infinity, II”, Bulletin of the University of Nizhny Novgorod, 2011, no. 1, 139–148 (in Russian)

[6] Dolov M. V., Chistyakova S. A., “On linear partial integrals of polynomial vector fields of the fourth degree with degenerate infinity, III”, Bulletin of the University of Nizhny Novgorod, 2011, no. 2, 123–129 (in Russian)

[7] Ushkho A. D., “Trajectories of cubic differential systems on a the plane with invariant lines of six various directions”, Proceedings of Voronezh State University. Ser. Physics. Mathematics, 2012, no. 2, 224–231 (in Russian)

[8] Tlyachev V. B., Ushkho A. D., Ushkho D. S., “Symmetry axes of planar polynomial differential systems”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 10:2 (2010), 41–49 (in Russian) | MR

[9] Ushkho D. S., “On straight isoclinal lines of cubic differential system”, Works of the Adygheya Republic Physical Society, 2003, no. 8, 7–21 (in Russian)

[10] Kurosh A. G., Higher algebra, Translated from the Russian by George Yankovsky; reprint of the 1972 translation, Mir, M., 1988, 428 pp.

[11] Lyubimova R. A., “On a differential equation with integral straight”, Differential and integral equations, 1, Izd-vo gos. un-ta, Gor'kii, 1977, 19–22 (in Russian)

[12] Llibre J., Vulpe N., “Planar Cubic Polynomial Differential Systems with the Maximum Number of Invariant Straight Lines”, Rocky Mountain J. Math., 36:4 (2006), 1301–1373 | DOI | MR | Zbl

[13] Llibre J., Vulpe N., “Cubic systems with invariant affine straight lines of total parallel multiplicity seven”, Electronic Journal of Differential Equations, 274 (2013), 1–22 (accessed 22, October, 2014) http://ejde.math.txstate.edu/ | MR

[14] (Accessed 22, October, 2014)

[15] Bujac C., “One new class of cubic systems with maximum number of invariant lines omitted in the classification of J. Llibre and N. Vulpe”, Buletinul Academiei de Stiinte a Republicii Moldova (BASM). Matematica, 2014, no. 2(75), 102–105 | Zbl