MINIMAL POLYNOMIAL BASIS OF CENTROAFFINE COMITANTS OF THE TWO-DIMENSIONAL SYSTEM OF
DIFFERENTIAL EQUATIONS WITH NONLINEARITIES OF THE FOURTH DEGREE
CIUBOTARU Stanislav
Institute of Mathematics and Computer Science of A.S.M.
Reviewer: CALIN IU., dr.
Keywords: system of differential equations, minimal polynomial basis, transvectant, comitants, invariants.
Let us consider the system of differential equations with nonlinearities of the fourth degree
𝑑𝑥 𝑗
𝑗
𝑗
= 𝑃1 (𝑥 1 , 𝑥 2 ) + 𝑃4 (𝑥 1 , 𝑥 2 ) (𝑗 = 1, 2)
(1)
𝑑𝑡
𝑗
𝑗
where 𝑃1 and 𝑃4 are homogeneous polynomials of degree 1 and 4, respectively, in 𝑥 1 and 𝑥 2 with real coefficients. We denote
by A the 14-dimensional coefficient space of system (1), by 𝑞 ∈ 𝑄 ⊆ 𝐴𝑓𝑓(2, ℝ) a linear transformation of the phase plane of (1)
and by 𝑟𝑞 (𝑎) its linear representation in the space A.
Definition 1. [1] A polynomial K(a, x) in coefficients of system (1) and coordinates of vectors 𝑥 ∈ ℝ2 is called a comitant of
system (1) with respect to a group Q if there exists a function 𝜆: 𝑄 ⟶ ℝ such that
𝐾(𝑟𝑞 (𝑎), 𝑞 ∙ 𝑥) ≡ 𝜆(𝑞)𝐾(𝑎, 𝑥)
for every 𝑞 ∈ 𝑄, 𝑎 ∈ 𝐴 and 𝑥 ∈ ℝ2 .
If Q is the group 𝐺𝐿(2, ℝ), then the comitant is called 𝐺𝐿(2, ℝ) - comitant or centroaffine comitant. In what follows only GLcomitants are considered. The function 𝜆(𝑞) is called a multiplicator. It is known [1] that the function 𝜆(𝑞) has the form
−𝑔
𝜆(𝑞) = ∆𝑞 , where ∆𝑞 = 𝑑𝑒𝑡(𝑞) and g is an integer, which is called the weight of the comitant K(a; x). If a comitant does not
depend on the coordinates of the vectors x, then it is called invariant.
We say that a homogeneous comitant K(a; x) has the character (r; g; d) if it has the weight g, the degree d with respect to the
coefficients of system (1) and the degree r with respect to the coordinates of the vector 𝑥 ∈ ℝ2.
Definition 2. [4] Let f and 𝜑 be polynomials in the coordinates of the vector 𝑥 = (𝑥 1 , 𝑥 2 ) ∈ ℝ2 of the degrees 𝑟 and 𝜌,
respectively. The polynomial
𝑘
(𝑓, 𝜑)
(𝑘)
(𝑟 − 𝑘)! (𝜌 − 𝑘)!
𝜕𝑘𝑓
𝜕𝑘 𝜑
𝑘
=
∑(−1)ℎ ( )
ℎ 𝜕(𝑥 1 )𝑘−ℎ 𝜕(𝑥 2 )ℎ 𝜕(𝑥 1 )ℎ 𝜕(𝑥 2 )𝑘−ℎ
𝑟! 𝜌!
ℎ=0
is called the transvectant of index k of polynomials f and 𝜑.
If polynomials f and 𝜑 are GL-comitants of system (1), then the transvectant of the index k ≤min(r; 𝜌) is a GL-comitant of the
system (1).
Definition 3. The set S of comitants (respectively, invariants) is called a polynomial basis of comitants (respectively, invariants)
for system (1) with respect to a group Q if any comitant (respectively, invariant) of system (1) with respect to the group Q can be
expressed in the form of a polynomial of elements of the set S.
Definition 4. A polynomial basis of comitants (respectively, invariants) for system (1) with respect to a group Q is called minimal
if by the removal from it of any comitant (respectively, invariant) it ceases to be a polynomial basis.
The theory of algebraic invariants and comitants for polynomial autonomous systems of differential equations has been
developed by C. Sibirschi [1, 2] and his disciples. One of the important problems concerning this theory is the construction of
minimal polynomial bases of the invariants and comitants of the mentioned systems, with respect to different subgroups of the
affine group of the transformations of their phase planes, in particular with respect to the subgroup GL(2;R). Some important
results in this direction are obtained by academician C. Sibirschi [1, 2] and N.Vulpe [3]. We remark, that polynomial bases for
𝑗
different particular combinations of homogeneous polynomials 𝑃𝑚 (𝑥 1 , 𝑥 2 ) (j = 1; 2; m = 0; 1; 2; 3) in system (1) are also
considered by E. Gasinskaya-Kirnitskaya, Dang Dinh Bich, D. Boularas, M. Popa, V. Ciobanu, V. Danilyuk, E. Naidenova.
We shall consider the following polynomials:
𝑅𝑖 = 𝑃𝑖1 𝑥 2 − 𝑃𝑖2 𝑥 1 , 𝑆𝑖 =
1 𝜕𝑃𝑖1 𝜕𝑃𝑖2
(
+
),
𝑖 𝜕𝑥 1 𝜕𝑥 2
(𝑖 = 1, 4)
(2)
which in fact are GL-comitants of the first degree with respect to the coefficients of system (1).
Using the comitants (2) as elementary "bricks" and the notion of transvectant we have constructed 409 irreducible GL-comitants
of system (1) and hence, the next result is proved:
Theorem 1. A minimal polynomial basis of 𝐺𝐿(2, ℝ)-comitants (respectively, of 𝐺𝐿(2, ℝ) −invariants) of system (1) up to 15
degree consists from 409 elements (respectively, 176 elements) which must be of the following 98 (respectively, 37) characters:
(2, -1, 1) – 1,
(3, 0, 2) – 2,
(5, -1, 1) – 1,
(5, -1, 2) -1,
(3, 0, 1) – 1,
(0, 0, 2) – 1,
(2, 2, 2) – 3,
(1, 4, 3) – 2,
(4, 1, 2) – 1,
(3, 3, 3) – 6,
(6, 0, 2) – 2,
(5, 2, 3) – 3,
(1, 1, 2) – 1,
(7, 1, 3) – 1,
(9, 0, 3) – 1,
(0, 6, 4) – 6,
(7, 1, 4) – 1,
(5, 5, 5) – 1,
(5, 2, 5) – 1,
(3, 6, 6) – 1,
(1, 10, 7) – 20,
(0, 6, 7) – 10,
(0, 9, 8) – 16,
(1, 10, 9) – 2,
(1, 10, 10) – 1,
(0, 9, 11) – 1,
(0, 18, 13) – 3,
(0, 3, 3) – 3,
(2, 5, 4) – 6,
(0, 3, 4) – 1,
(7, 4, 5) – 1,
(0, 3, 5) – 3,
(5, 5, 6) – 1,
(3, 9, 7) – 1,
(1, 4, 7) – 2,
(1, 7, 8) – 2,
(0, 9, 9) – 5,
(0, 9, 10) – 3,
(0, 6, 11) – 1,
(0, 15, 13) – 1,
(2, 2, 3) – 4,
(4, 4, 4) – 6,
(2, 2, 4) – 3,
(0, 6, 5) – 6,
(2, 2, 5) – 2,
(0, 6, 6) – 8,
(5, 8, 7) – 1,
(0, 3, 7) – 1,
(0, 6, 8) – 5,
(1, 7, 9) – 1,
(0, 6, 10) – 1,
(0, 18, 12) – 4,
(0, 12, 13) – 1,
(4, 1, 3) – 3,
(6, 3, 4) – 2,
(4, 1, 4) – 1,
(2, 5, 5) – 9,
(0, 9, 6) – 7,
(2, 5, 6) – 5,
(0, 9, 7) – 15,
(0, 12, 8) – 15,
(1, 4, 8) – 1,
(0, 6, 9) – 3,
(1, 16, 11) – 2,
(1, 16, 12) – 1,
(0, 18, 14) – 1,
(6, 0, 3) – 1,
(1, 4, 4) – 8,
(1, 1, 4) – 1,
(4, 4, 5) – 3,
(2, 8, 6) – 12,
(1, 4, 6) – 4,
(2, 8, 7) – 5,
(2, 11, 8) – 4,
(1, 13, 9) – 5,
(0, 15, 10) – 14,
(0, 15, 11) – 5,
(0, 15, 12) – 3,
(0, 15, 14) – 1,
(1, 1, 3) – 2,
(3, 0, 3) – 1,
(3, 3, 4) – 6,
(5, 2, 4) – 1,
(1, 7, 5) – 11,
(3, 6, 5) – 9,
(1, 4, 5) – 9,
(3, 3, 5) – 1,
(4, 7, 6) – 2,
(1, 7, 6) – 20,
(3, 3, 6) – 1,
(0, 3, 6) – 1,
(1, 7, 7) – 7,
(3, 6, 7) – 1,
(1, 10, 8) – 7,
(3, 9, 8) – 1,
(3, 12, 9) – 1,
(0, 12, 9) – 19,
(1, 13, 10) – 2, (0, 12, 10) – 5,
(1, 13, 11) – 1, (0, 12, 11) – 3,
(0, 12, 12) – 1, (0, 9, 12) – 1,
(0, 21, 15) – 1, (0, 18, 15) – 1.
Remark. Besides each of the characters given in theorem above, we have also indicated the number of the irreducible
comitants (or invariants) having this character.
References:
1. K. S. Sibirsky, Introduction to the Algebraic Theory of Invariants of Differential Equations. Manchester University Press,
1988.
2. K. S. Sibirsky, Algebraic invariants of differential equations and matrices. Kishinev, Shtiintsa, 1976, (in Russian).
3. N.I. Vulpe, Polynomial bases of comitants of differential systems and their applications in qualitative theory. Kishinev,
Shtiintsa, 1986, (in Russian).
4. G. B. Gurevich, Foundations of the Theory of Algebraic Invariants. Noordhoff, Groningen, 1964.