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
(𝑗 = 1, 2),
= 𝑃1 (𝑥 1 , 𝑥 2 ) + 𝑃4 (𝑥 1 , 𝑥 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 say that a homogeneous 𝐺𝐿(2, ℝ)-comitant (centroaffine 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 [1].
Definition 1. 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 2. 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 GLcomitants of system (1) and hence, the next result is proved:
Theorem: 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.
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).