CUBIC SYSTEMS WITH A REAL INVARIANT STRAIGHT LINE OF MAXIMAL INFINITESIMAL
MULTIPLICITY
VACARAŞ Olga
Institute of Mathematics and Computer Science A.S.M., Republic of Moldova
Reviewer: PhD A.ŞUBĂ
Keywords: system of differential equations, invariant straight line, infinitesimal multiplicity
We consider the real polynomial system of differential equations
x  Px; y ; y  Qx; y ; GCD P; Q  1
and the vector field X  Px; y 
(1)


associate to system (1).
 Qx; y 
x
y
Denote
n  max degP , degQ . If n  3 then system (1) is called cubic. An algebraic curve f x; y   0,
f  Cx; y  is called invariant algebraic curve of the system (1) if there exists a polynomial K f  Cx; y  such that the
identity X f   f x; y  K f x; y ,
x; y  R 2 holds. In particular, a straight line  x   y    0,  ,  ,   C,     0
invariant for (1) if there exist a polynomial K x; y  such that the identity Px; y   Qx; y   x  y   K x; y  holds.
is
In the work [1] there are introduced the following definitions of the multiplicity of an invariant algebraic curve:
algebraic multiplicity, integrable multiplicity, infinitesimal multiplicity, geometric multiplicity, holonomic multiplicity and the
relations between these definitions are established.
Definition 1. Let f  0 be an invariant algebraic curve of degree d of a polynomial vector field X of degree n . We
say that
F  f 0  f 1  ... f k 1 k 1  Cx, y,   /  k
defines a generalized invariant algebraic curve of order k based on f  0 if f 0  f ,..., f k 1 are polynomials in C[ x, y] of
degree at most d , and F satisfies the equation X F   FL F , for some polynomial
 
 
LF  L0  L1  ... Lk 1 k 1  Cx, y,   /  k
which must necessarily be of degree at most n  1 in x and y . We call L F the cofactor of F . Equivalently, the equation
X F   FL F can be written as
(2)
X  f i   f i L0  f i 1 L1  ...  f 0 Li , i  0,..., k  1.
Definition 2. Let f  0 be an invariant algebraic curve of degree d in a polynomial vector field X of degree n .
We say that f  0 is of infinitesimal multiplicity m with respect to X if m is the maximal order of all nondegenerate
generalized invariant algebraic curves of X based on f . If no such maximum exists, then the infinitesimal multiplicity is
said to be infinite.
Theorem. For cubic systems the infinitesimal multiplicity of a real invariant straight line is at most seven.
Any cubic system having an invariant straight line of the infinitesimal multiplicity seven can be brought to the form
(3)
x  x 3 , y  1  ax 2  bx 3  3x 2 y.
For the system (3) only the straight line x  0 is invariant and we have:
f 0  x; f 1  x 1   1 ; f 2  x 2   2 ; f 3  x 3  y 13   3 ; f 4  x 4  2 y 1 13  3 y 12  2   4 ; f 5  x 5  3 y 12  13  2 y 2  13 
 ay 15  6 y 1 12  2  3 y 1 22  3 y 12  3   5 ; f 6  x 6  4 y 13  13  6 y 1 2  13  2 y 3  13  4ay 1 15  by 16  9 y 12  12  2 
 6 y 2  12  2  5ay 14  2  6 y 1 1 22  y 23  6 y 1 12  3  6 y 1 2  3  3 y 12  4   6 ,  j ,  j  are
parameters.
The equality (2) gives us the polynomials L j , j  0,1,..., 6. For example,
L 0  x ; L 1   x 1 ; L 2  x 1 1   12  x 2 ; L 3  x 12  1  x 2  1  2 1 12  ax 13  bx 2  13  2 xy 13  x 1 2  2 1 2  x 3 ;
2
L 4  x 13  1  2 x 1 2  1  x 3  1  3 12  12  2 2  12  3ax 1 13  3bx 2 1 13  6 xy 1 13  a 14  bx 14  y 14  x 12  2  x 2  2 
 4 1 1 2  3ax 12  2  3bx 2  12  2  6 xy 12  2   22  x 1 3  2 1 3  x 4 .
References
1. C. Christopher, J. Llibre and J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields,
Pacific Journal of Mathematics, 329 (2007), no. 1, 63-117.