First Integrals and Invariants of Systems of Ordinary Differential Equations

Abstract

We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincaré–Dulac normal forms for autonomous systems of ordinary differential equations with a diagonal matrix of the linear part. Using tools from computational algebra, we develop an algorithmic approach for identifying generators of the algebras of monomial and polynomial first integrals, which works in the general case where the matrix of the linear part includes algebraic complex eigenvalues. Our method also provides a practical tool for exploring the algebraic structure of polynomial invariants and their relation to the Poincaré-Dulac normal forms of the underlying vector fields.

1. Introduction

First integrals and invariants are fundamental tools in the study of systems of ordinary differential equations (ODEs). They allow us to better understand the qualitative behavior of solutions by revealing deep structural properties of the system. In particular, first integrals—functions that remain constant along solution trajectories—help identify conserved quantities such as energy, momentum, or other geometric invariants. These conserved quantities can simplify the analysis of a system, reduce the number of effective variables, or even allow the system to be integrated completely in special cases (see e.g., [1,2,3,4,5,6,7,8] and the references given there). Invariants, in general, provide information about the symmetries and stability of the system. The presence of invariants is often key to the classification of differential systems up to certain equivalence relations.

The connection between monomial first integrals and Poincaré-Dulac normal forms has been studied extensively in [6,9,10,11]. The theory of polynomial invariants for parametric families of ODEs was developed by Sibirsky and his school [12,13]. These invariants are important for the classification of ODEs [14] and for studies related to the center-focus problem [15,16,17].

In this paper, we study monomial first integrals, polynomial invariants, and their connection to Poincaré-Dulac normal forms for the n-dimensional autonomous system

$$\dot{x}=Ax+X\left(x\right),x\in {\mathbb{C}}^n,$$

(1)

where A is a complex diagonal matrix,

$$A=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n),$$

(2)

and $X\left(x\right)$ is a power series without constants or linear terms.

Let ${\mathcal{V}}^n$ be the space of polynomial vector fields $v:{\mathbb{C}}^n\to {\mathbb{C}}^n$ and, for $j>1$, let ${\mathcal{V}}_j^n\subseteq {\mathcal{V}}^n$ be the subspace of vector fields v whose components $v_i$ are homogeneous polynomials of degree j, for $i=1,\dots ,n$. It is not difficult to see that any formal invertible change of coordinates of the form

$$x=y+H\left(y\right)=y+\sum _{j=2}^{\infty }{H}_j\left(y\right),$$

(3)

with $H_j\in {\mathcal{V}}_j^n$ for all $j\ge 2$, brings system (vector field) (1) to a system of a similar form,

$$\dot{y}=Ay+G\left(y\right),$$

(4)

where

$$G\left(y\right)=\sum _{j=2}^{\infty }{G}_j\left(y\right),\mathrm{and} G_j\left(y\right)\in {\mathcal{V}}_j^n\mathrm{for} \mathrm{all} j\ge 2.$$

Let ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ be the set of non-negative integers. For $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0^n$, we define $x^{\alpha }:={\prod }_{i=1}^n{x}_i^{{\alpha }_i}$, where $x=(x_1,\dots ,x_n)$ is a column vector. Let $e_k$ be the n-dimensional unit vector for $k=1,\dots ,n$. A term $a{y}^{\alpha }$ in $e_{k}G\left(y\right)$ or $b{x}^{\alpha }$ in $e_{k}X\left(x\right)$ is called resonant if

$$\langle \lambda ,\alpha \rangle -{\lambda }_k=0,$$

(5)

where $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ and $\langle \lambda ,\alpha \rangle$ is the usual inner product of n-tuples. We denote by ${\mathcal{N}}_{\lambda }^{\left(k\right)}$ the set of all solutions $\alpha \in {\mathbb{N}}_0^n$ to (5), that is

$${\mathcal{N}}_{\lambda }^{\left(k\right)}=\{\alpha \in {\mathbb{N}}_0^n:\langle \lambda ,\alpha \rangle -{\lambda }_k=0\}.$$

(6)

System (4) is said to be in the Poincaré–Dulac normal form, or simply in normal form, if $G\left(y\right)$ contains only resonant terms. By the Poincaré-Dulac theorem [18,19], any system (1) can be transformed to a normal form by a suitable transformation (3) [18,19,20]. In particular, transformation (3), which brings system (1) to a Poincaré–Dulac normal form, is called a normalization or a normalizing transformation.

Equivalently (see e.g., [6,11,21]), system (4) is in the Poincaré–Dulac normal form if $[Ay,G(y\left)\right]=0,$ that is, for all $j\ge 2$, $[Ay,G_j\left(y\right)]=0.$

Following [9], we define the centralizers

$$C_0^{for}\left(A\right):=\{g\in \mathbb{C}{\left[[x_1,\dots ,x_n]\right]}^n:[g,A]=0\}$$

(7)

and

$$C^{pol}\left(A\right):=\{g\in \mathbb{C}{[x_1,\dots ,x_n]}^n:[g,A]=0\}.$$

By Lemma 2.9 of [9] and Corollary 4.5.9 of [22], $C_0^{for}\left(A\right)$ is spanned as a $\mathbb{C}$-vector space by all $x^{\alpha }{e}_k$ with $\alpha \in {\mathcal{N}}_{\lambda }^{\left(k\right)}$. To our knowledge, at present there are no algorithmic methods available to compute a generating set of $C_0^{for}\left(A\right)$ for a given A. However, some particular cases are discussed in [9,22].

Recall that a polynomial $f\in k[x_1,\dots ,x_n]$ (where k is a field) is invariant under the action of the group $\mathfrak{G}$ (or simply an invariant of $\mathfrak{G}$) if $f\left(x\right)=f\left(\mathfrak{g}x\right)$ for every $x\in k^n$ and every $\mathfrak{g}\in \mathfrak{G}$.

In this paper, we first deal with monomial first integrals of the linear approximation of (1), that is, of the linear system

$$\dot{x}=Ax,$$

(8)

where A is a complex diagonal matrix (2), and propose an algorithm to compute a minimal generating set of the algebra of monomial first integrals of system (8). Subsequently, we establish a connection between specific monomial invariants of the polynomial system (1) and first integrals of the linear differential system (2). Furthermore, we adapt our algorithm to compute a generating set of the algebra of invariants for system (1), applicable even when some eigenvalues are not rational but algebraic elements. In Section 5, we demonstrate how our algorithm effectively describes the structures of $C_0^{for}\left(A\right)$ and $C^{pol}\left(A\right)$.

This work contributes to the broader understanding of polynomial invariants, monomial first integrals, and the structure of Poincaré-Dulac normal forms of ordinary differential equations.

2. Preliminaries

For system (8) where $A=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, denote

$${\mathcal{M}}_{\lambda }:=\left\{\alpha \in {\mathbb{N}}_0^n|\langle \lambda ,\alpha \rangle =0\right\}=\left\{\alpha \in {\mathbb{N}}_0^n|A{\alpha }^{\top }=0\right\}.$$

(9)

Clearly, ${\mathcal{M}}_{\lambda }$ is a submonoid of the additive monoid ${\mathbb{Z}}^n$. It is obvious that the elements $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ of ${\mathbb{N}}_0^n$ are in one-to-one correspondence with the monic monomials $x^{\alpha }=x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}$ of the polynomial ring $\mathbb{C}[x_1,\dots ,x_n]$. Moreover, since the derivative of $x^{\alpha }$ along the vector field of the system $\dot{x}=\mathrm{diag}({\lambda }_1,...,{\lambda }_n)x$ is

$${\alpha }_1{\displaystyle \frac{x^{\alpha }}{x_1}}{\lambda }_1{x}_1+\cdots +{\alpha }_n{\displaystyle \frac{x^{\alpha }}{x_n}}{\lambda }_n{x}_n=\langle \alpha ,\lambda \rangle x^{\alpha }$$

(10)

we see that $x^{\alpha }$ is a first integral of (8) if and only if $\langle \alpha ,\lambda \rangle =0.$ Hence, the monoid ${\mathcal{M}}_{\lambda }$ encodes precisely the exponent vectors of all monomial first integrals of the linear system (8). This correspondence forms the algebraic foundation for describing the structure of polynomial invariants associated with the system.

Note that since we are studying only monomial first integrals of the linear system (8) with the diagonal matrix, there are no interesting dynamics involved. Of course, such integrals represent conservation laws, and from a geometrical point of view, each such integral defines a union of hyperplanes. However, as we will see below, such integrals play an important role from an algebraic perspective, as they allow us to describe the structure of the normal form module of the nonlinear system (1).

Let $I\left(\lambda \right)$ be the algebra of polynomial first integrals of (8) and ${\mathcal{R}}_{\lambda }$ be the $\mathbb{Z}$-module spanned by the elements of ${\mathcal{M}}_{\lambda }$. By the results of [6,11], we have the following statement.

Lemma 1

(Lemma 2 of [6]). The algebra $I\left(\lambda \right)$ of polynomial first integrals of system (8) is finitely generated $\mathbb{C}$-algebra. Furthermore, if $\mathit{rank}\left({\mathcal{R}}_{\lambda }\right)=d$, then there are exactly d functionally independent polynomial first integrals of (8).

Lemma 2.

The monoid ${\mathcal{M}}_{\lambda }$ has a unique generating set $H_{\lambda }$ (called the Hilbert basis of ${\mathcal{M}}_{\lambda }$).

Proof. 

By Lemma 1, the algebra of polynomial first integrals of (8) is finitely generated, so the additive monoid ${\mathcal{M}}_{\lambda }$ is also finitely generated. As a submonoid of ${\mathbb{Z}}^n$ with only one unit element, namely $\overline{0}=(0,\dots ,0)$, ${\mathcal{M}}_{\lambda }$ is the so-called pointed affine monoid (see [23] for definition). By Proposition 7.15 of [23], it has a unique minimal finite generating set.    □

The vector monomials $x^{\alpha }{e}_k,\alpha \in {\mathcal{N}}_{\lambda }^{\left(k\right)}$, and the elements of the $\mathbb{C}$-vector space they generate are often called equivariants. Note that here we adopt the terminology of book [22] and these equivariants have nothing in common with the so-called ${\mathbb{Z}}_q$-equivariant considered e.g., in [24]. By Proposition 1.6 of [11] (see also Lemma 4.2 of [9]), $C^{pol}\left(A\right)$ is a finitely generated $I\left(\lambda \right)$-module. Thus there arises an important problem to study the structure of $C^{pol}\left(A\right)$ (see e.g., [9,11,22]). By Lemma 4.4 of [9], if ${\lambda }_i\ne 0$ $(i=1,\dots ,n)$ and all $\alpha \in {\mathcal{N}}_{\lambda }^{\left(k\right)}$ ($k=1,\dots ,n$) have non-negative entries, then $C^{pol}\left(A\right)$ is a free $I\left(\lambda \right)$-module of rank n generated by $v_j\left(x\right)=x_j{e}_j$ where $j=1,\dots ,n$. However, in general, the structure of $C^{pol}\left(A\right)$ can be rather complicated, and, to describe it, one needs to find generators for ${\mathcal{M}}_{\lambda }$ and describe the sets ${\mathcal{N}}_{\lambda }^{\left(k\right)}$, $k=1,\dots ,n.$ We consider this problem in Section 5. In particular, in this section, we present an algorithm to compute the equivariants of system (8) and a unique presentation of $C^{pol}\left(A\right)$.

In the next section, we present algorithms to compute a Hilbert basis of ${\mathcal{M}}_{\lambda }$, and hence a generating set of $I\left(\lambda \right)$, as well as an algorithm to compute a generating set of the $\mathbb{Z}$-module ${\mathcal{R}}_{\lambda }$.

3. Monomial First Integrals

Obviously, a monomial $x^{\alpha }$, $\alpha \in {\mathbb{Z}}^n$, of the ring of Laurent polynomials in $x_1,\dots ,x_n$, is a first integral of linear system (8) if and only if

$$\langle \alpha ,\lambda \rangle =0.$$

(11)

Such a monomial $x^{\alpha }$ is called a Laurent monomial first integral. The monomials corresponding to solutions $\alpha \in {\mathbb{N}}_0^n$ of equation (11) generate the algebra of polynomial first integrals $I\left(\lambda \right)$ of (8), which is finitely generated by Lemma 1. Since a polynomial $r\left(x\right)$ is a first integral of (8) if and only if each of its monomials is a first integral of (8), to describe the algebra $I\left(\lambda \right)$, it is sufficient to find the monomial first integrals of (8). By Lemma 2, the algebra $I\left(\lambda \right)$ has a unique generating set. In this section, we address the problem of determining generators of the algebra under certain assumptions on $\lambda$.

Let ${\lambda }_1,\dots ,{\lambda }_n$ be algebraic elements of $\mathbb{C}$, that is, each ${\lambda }_i$ is a root of a polynomial with integer coefficients. Let $K=\mathbb{Q}({\lambda }_1,\dots ,{\lambda }_n)$ be the finite algebraic extension of $\mathbb{Q}$ containing ${\lambda }_1,\dots ,{\lambda }_n$. Let $b_1,\dots ,b_d$ be a basis of K over $\mathbb{Q}$. Then, each ${\lambda }_i$ is a $\mathbb{Q}$-linear combination of $b_j$’s. That is,

$${\lambda }_i=\sum _{j=1}^d{c}_{ji}{b}_j$$

(12)

where $c_{ji}\in \mathbb{Q}$. Let $C=\left[c_{ji}\right]$ be the $d\times n$ matrix whose $ji$-entry is $c_{ji}$ and $l=lcm(c_{ji}:j=1,\dots ,d \mathrm{and} i=1,\dots ,n)$. Let $\mathfrak{A}=lC=\left[l{c}_{ji}\right]$ be the integer $d\times n$-matrix obtained from C by clearing the denominators. In particular,

$$\mathfrak{A}=[{\mathfrak{a}}_1{\mathfrak{a}}_2\cdots {\mathfrak{a}}_n],$$

(13)

where ${\mathfrak{a}}_i$ represents the i-th column of the matrix and corresponds to ${\lambda }_i$.

Lemma 3.

For $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\in {\mathbb{Z}}^n$,

$$\langle \alpha ,\lambda \rangle =\sum _{i=1}^n{\lambda }_i{\alpha }_i=0⟺\mathfrak{A}\alpha =0.$$

Proof. 

We use the notations presented in the paragraph preceding the lemma. Using the basis $\{b_1,\dots ,b_d\}$ of K over $\mathbb{Q}$ and the expressions (12), we have $\langle \alpha ,\lambda \rangle ={\sum }_{j=1}^d\left({\sum }_{i=1}^n{c}_{ji}{\alpha }_i\right)b_j$. Since $b_1,\dots ,b_d$ are independent over $\mathbb{Q}$, it yields that $\langle \alpha ,\lambda \rangle =0$ if and only if

$$\sum _{i=1}^n{c}_{ji}{\alpha }_i=0\mathrm{for}\mathrm{all}j=1,\dots ,d,$$

which is equivalent to $C{\alpha }^{\top }=0$ and also $\mathfrak{A}{\alpha }^{\top }=lC{\alpha }^{\top }=0$.    □

By Lemma 3, the set $\left\{\alpha \in {\mathbb{N}}_0^n|\mathfrak{A}{\alpha }^{\top }=0\right\}$ is the same as the monoid ${\mathcal{M}}_{\lambda }$ defined by (9).

Notice that

$$K_{\rho }=\left\{\alpha \in {\mathbb{Z}}^n|\mathfrak{A}{\alpha }^{\top }=0\right\}$$

(14)

is a subgroup of the finitely generated abelian group ${\mathbb{Z}}^n$. Therefore, $K_{\rho }$ is itself a finitely generated abelian group, and its basis can be readily computed, for example, using Smith normal form, as the kernel of the group homomorphism

$$\rho :{\mathbb{Z}}^n⟶{\mathbb{Z}}^d,\rho \left(\nu \right)=\mathfrak{A}\nu .$$

Furthermore, the monoid

$${\mathcal{M}}_{\lambda }=\left\{\alpha \in {\mathbb{N}}_0^n|\mathfrak{A}{\alpha }^{\top }=0\right\}$$

(15)

consists of the non-negative integer solutions in the kernel. However, computing the Hilbert basis $H_{\lambda }$ of ${\mathcal{M}}_{\lambda }$ and hence a minimal generating set of $I\left(\lambda \right)$ is subtle and generally requires tools from integer linear programming or commutative algebra, such as Gröbner bases or Hilbert basis techniques [25,26,27].

Algorithm 1.4.5 of [28] employs Gröbner basis techniques to compute the Hilbert basis of the monoid ${\mathcal{M}}_{\lambda }$ associated with the matrix $\mathfrak{A}$. Next, we present Algorithm 1, which is essentially the same procedure but augmented with the additional steps detailed above for constructing the matrix $\mathfrak{A}$ corresponding to a given vector $\lambda$. The correctness of Algorithm 1 follows from the correctness of Algorithm 1.4.5 of [28], together with Equation (10).

The bottleneck in the computation of Algorithm 1 lies in the associated Gröbner basis calculation. The complexity of Gröbner basis computations has been studied extensively. For further details, we refer the reader to [29,30,31].

Example 1.

Let $\lambda =(1,\zeta ,{\zeta }^2,-2,3)$ where ζ is the third root of unity. The corresponding system (8) is

$$\dot{x}=\mathrm{diag}(1,\zeta ,{\zeta }^2,-2,3)x.$$

(16)

Note that, ${\zeta }^2=-\zeta -1$, and hence, $\mathbb{Q}\left(\lambda \right)=\mathbb{Q}\left(\zeta \right)$, and $\{1,\zeta \}$ is a basis for $\mathbb{Q}\left(\zeta \right)$ over $\mathbb{Q}$. Therefore,

$$\begin{array}{ccc}\hfill {\lambda }_1& =& 1=1·1+0·\zeta \hfill \\ \hfill {\lambda }_2& =& \zeta =0·1+1·\zeta \hfill \\ \hfill {\lambda }_3& =& {\zeta }^2=-1-\zeta =-1·1+(-1)·\zeta \hfill \\ \hfill {\lambda }_4& =& -2=2·1+0·\zeta \hfill \\ \hfill {\lambda }_5& =& 3=3·1+0·\zeta .\hfill \end{array}$$

Hence,

$$\mathfrak{A}=\left[\begin{array}{ccccc}1& 0& -1& -2& 3\\ 0& 1& -1& 0& 0\end{array}\right].$$

Implementing Algorithm 1 in the computer algebra system Macaulay2 [32], we obtain the Hilbert basis of ${\mathcal{M}}_{\lambda }$ to be

$$H_{\lambda }=\{(0,0,0,3,2),(0,1,1,1,1),(0,3,3,0,1),(1,0,0,2,1),(1,1,1,0,0),(2,0,0,1,0)\}$$

and the generators of $I\left(\lambda \right)$ are, therefore, equal to

$$I_1=x_4^3{x}_5^2,I_2=x_2{x}_3{x}_4{x}_5,I_3=x_2^3{x}_3^3{x}_5,I_4=x_1{x}_4^2{x}_5,I_5=x_1{x}_2{x}_3,I_6=x_1^2{x}_4.$$

(17)

The Macaulay2 code and calculations for the example are given in Appendix A.

Algorithm 1: Hilbert basis of ${\mathcal{M}}_{\lambda }$ and generators of $I\left(\lambda \right)$

Input: A vector $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{C}}^n$ of algebraic elements Output: Hilbert basis $H_{\lambda }$ of ${\mathcal{M}}_{\lambda }$ and a minimal generating set of $I\left(\lambda \right)$ 1  Find a number field K containing all ${\lambda }_i$ 2  Choose a $\mathbb{Q}-$basis $\{b_1,\dots ,b_d\}$ for K 3  Express each ${\lambda }_i$ in the chosen basis: Write $${\lambda }_i=c_{1i}{b}_1+c_{2i}{b}_2+\cdots +c_{di}{b}_d,\mathrm{where} c_{ji}\in \mathbb{Q}$$ 4  Let $C=\left[c_{ji}\right]$ be the $d\times n$-matrix of coefficients 5  Convert to integer matrix: Let $\mathfrak{A}=\left[a_{ji}\right]=lC=\left[l{c}_{ji}\right]$ be the integer $d\times n$-matrix obtained from C by clearing the denominators 6  Compute the reduced Gröbner basis $\mathcal{G}$ for the ideal $$〈{\displaystyle x_i-z_i\prod _{j=1}^d{t}_j^{a_{ji}}:i=1,\dots ,n}〉$$ in $k[t_1,\dots ,t_d,t_1^{-1},\dots ,t_d^{-1},x_1,\dots ,x_n,z_1,\dots ,z_n]$ with respect to ≻: $$\{t_1^{\pm },\dots ,t_d^{\pm }\}\succ \{x_1,\dots ,x_n\}\succ \{z_1,\dots ,z_n\}.$$ 7  The Hilbert basis $H_{\lambda }$ of ${\mathcal{M}}_{\lambda }$ consists of all vectors $\nu$ such that ${\mathbf{x}}^{\nu }-{\mathbf{z}}^{\nu }$ appears in $\mathcal{G}$. 8  The monomials $x^{\nu }$ such that ${\mathbf{x}}^{\nu }-{\mathbf{z}}^{\nu }$ appears in $\mathcal{G}$ generate $I\left(\lambda \right)$.

We note that the following statement takes place.

Proposition 1.

If the rank of $ker\rho$ is equal to p and the rank of ${\mathcal{R}}_{\lambda }$ is also p, then, both $K_{\rho }$ and ${\mathcal{R}}_{\lambda }$ are generated by p linearly independent elements of $H_{\lambda }$.

It is worth noting here that ${\mathcal{R}}_{\lambda }$ may not be free, as $\mathbb{Z}-$syzygies may exist among the elements of ${\mathcal{H}}_{\lambda }$. To compute a minimal generating set for the $\mathbb{Z}$-module ${\mathcal{R}}_{\lambda }$, classical methods from linear algebra can be used, as in Algorithm 2 presented below. The correctness of this algorithm is straightforward and follows directly from the fact that Gaussian elimination produces a basis for the column space.

Algorithm 2: Generating Set of the $\mathbb{Z}$-Module ${\mathcal{R}}_{\lambda }$

Input: Hilbert basis $H_{\lambda }$ Output: Generating set of the $\mathbb{Z}$-module ${\mathcal{R}}_{\lambda }$ 1  Let N be the matrix whose columns are the vectors in the Hilbert basis $H_{\lambda }$. 2  Compute the column space of N over $\mathbb{Z}$ using Gaussian elimination. 3  The set of columns of N corresponding to pivot columns in the reduced echelon form forms a generating set for the $\mathbb{Z}$-module ${\mathcal{R}}_{\lambda }$.

In Example 1, the matrix

$$N=\left[\begin{array}{cccccc}0& 0& 0& 1& 1& 2\\ 0& 1& 3& 0& 1& 0\\ 0& 1& 3& 0& 1& 0\\ 3& 1& 0& 2& 0& 1\\ 2& 1& 1& 1& 0& 0\end{array}\right],$$

obtained from the Hilbert basis, after the Gaussian elimination, is reduced to

$$\left[\begin{array}{cccccc}1& 0& -1& 0& -1& -1\\ 0& 1& 3& 0& 1& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right].$$

Notice that columns 1, 2, and 4 have the leading terms and hence the $\mathbb{Z}$-module ${\mathcal{R}}_{\lambda }$ is of rank 3 and is generated by

$$\left\{\right(0,0,0,3,2),(0,1,1,1,1),(1,0,0,2,1\left)\right\}.$$

Since the number of generators of ${\mathcal{R}}_{\lambda }$ is 3, system (16) has three functionally independent polynomial first integrals, which can be chosen to be $I_1,I_2,I_4$. The calculation of the reduced row echelon form of matrix N was performed in SAGE [33], see Appendix B.

4. Invariants

In this section, we study polynomial invariants of n-dimensional polynomial systems of ODEs under the action of a one-parameter group on the phase space of the system.

Let ${\mathbf{N}}_i=\{Q=(q_1,\dots ,q_n)\in {\mathbb{Z}}^n:q_i\ge -1,q_j\ge 0\mathrm{if}j\ne i\}$ and $\mathbf{N}={\mathbf{N}}_1\cup \cdots \cup {\mathbf{N}}_n$. Following Bruno [4], we can write any n-dimensional analytical or formal system of ODEs in the form

$$\dot{x}=\sum _{Q\in \mathbf{N}}(x\odot a_Q)x^Q,$$

(18)

where the ⊙ stands for the Hadamard product,

$$a_Q={(a_1^{\left(Q\right)},\dots ,a_n^{\left(Q\right)})}^{\top }\in {\mathbb{C}}^n.$$

(19)

More specifically, any n-dimensional polynomial system can be written as

$$\dot{x}=\sum _{Q\in \Omega }(x\odot a_Q)x^Q,$$

(20)

where $\Omega$ is a finite set, say

$$\Omega =\{Q_1,\dots ,Q_{\mathsf{\ell }}\}\subseteq \mathbf{N}.$$

The matrix $A=\mathrm{diag}\lambda ,$ where $\lambda =\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right)$ is the infinitesimal generator of the complex matrix group $e^{A\varphi }$. After the change of variables

$$y=e^{A\varphi }x,$$

(21)

we obtain from (20) the system

$$\dot{y}=y\odot \sum _Q{a}_Q{e}^{-\langle \lambda ,Q\rangle \varphi }{y}^Q=y\odot \sum _Q{\widehat{a}}_Q{y}^Q.$$

In particular, the coefficients are changed by the rule

$$a_Q\mapsto {\widehat{a}}_Q=a_Q{e}^{-\langle \lambda ,Q\rangle \varphi },$$

(22)

where, as above, $\langle \lambda ,Q\rangle$ represents the usual inner product of n-tuples $\lambda$ and Q.

Let $\mathfrak{m}=n\mathsf{\ell }$ be the number of parameters $a_i^{\left(Q\right)}$ in (20). Define the ordered $\mathfrak{m}$-tuple of parameters of system (20) as

$$a=\left(a_{Q_1}^{\top },\dots ,a_{Q_{\mathsf{\ell }}}^{\top }\right)=\left(a_1^{\left(Q_1\right)},\dots ,a_n^{\left(Q_1\right)},\dots ,a_1^{\left(Q_{\mathsf{\ell }}\right)},\dots ,a_n^{\left(Q_{\mathsf{\ell }}\right)}\right)$$

(23)

and consider the algebra of complex polynomials $\mathbb{C}\left[a\right]$, where the variables are the parameters of system (20). Let

$$\nu \left(i\right)=({\nu }_1^{\left(i\right)},\dots ,{\nu }_n^{\left(i\right)}),i=1,\dots ,\mathsf{\ell }$$

and

$$\nu =\left(\nu \right(1),\dots ,\nu (\mathsf{\ell }\left)\right).$$

For $a_Q$ defined by (19) and $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$, let

$${\left(a^{\left(Q\right)}\right)}^{\alpha }:={\left(a_1^{\left(Q\right)}\right)}^{{\alpha }_1}{\left(a_2^{\left(Q\right)}\right)}^{{\alpha }_2}\dots {\left(a_n^{\left(Q\right)}\right)}^{{\alpha }_n}.$$

Given the ordered $\mathfrak{m}$-tuple a, the monomial $a^{\nu }$ in $\mathbb{C}\left[a\right]$ is defined by

$$a^{\nu }=\prod _{i=1}^{\mathsf{\ell }}{a}_{Q_i}^{\nu \left(i\right)}.$$

The change of variables (21) (the group action (22)) induces a $\mathbb{C}$-linear map on $\mathbb{C}\left[a\right]$ acting on monomial $a^{\nu }$ as

$$a^{\nu }\mapsto \prod _{i=1}^{\mathsf{\ell }}{\widehat{a}}_{Q_i}^{\nu \left(i\right)}=\prod _{i=1}^{\mathsf{\ell }}{a}_{Q_i}^{\nu \left(i\right)}{e}^{-{\sum }_{k=1}^n\langle \lambda ,Q_i\rangle {\nu }_k^{\left(i\right)}}=\prod _{i=1}^{\mathsf{\ell }}{a}_{Q_i}^{\nu \left(i\right)}{e}^{-\langle \lambda ,{\mathcal{Q}}_i\nu {\left(i\right)}^{\top }\rangle },$$

(24)

where ${\mathcal{Q}}_i$ is $n\times n$ matrix whose each column is $Q_i.$ Let $\mathcal{Q}$ be the $n\times \mathfrak{m}$ matrix

$$\mathcal{Q}=[{\mathcal{Q}}_1{\mathcal{Q}}_2\cdots {\mathcal{Q}}_{\mathsf{\ell }}]$$

and let

$$L:{\mathbb{N}}_0^{\mathfrak{m}}\to {\mathbb{C}}^n$$

be the map

$$L\left(\nu \right):={\left(L^1\left(\nu \right),L^2\left(\nu \right),\dots ,L^n\left(\nu \right)\right)}^{\top }=\mathcal{Q}{\nu }^{\top }.$$

(25)

Using the additive map (25), we define the monoid

$${\mathcal{M}}_L=\left\{\nu \in {\mathbb{N}}_0^{\mathfrak{m}}:\langle \lambda ,L\left(\nu \right)\rangle =0\right\},$$

(26)

which is finitely generated by the same argument used for ${\mathcal{M}}_{\lambda }$ introduced above.

Theorem 1.

A monomial $a^{\nu }$ is an invariant of group (22) if and only if $\nu \in {\mathcal{M}}_L$.

Proof. 

From (24), we see that the monomial $a^{\nu }$ is an invariant of (22) if and only if

$$\sum _{i=1}^{\mathsf{\ell }}\langle \lambda ,{\mathcal{Q}}_i\nu {\left(i\right)}^{\top }\rangle =0.$$

But, the above equality is equivalent to

$$\sum _{i=1}^{\mathsf{\ell }}\langle \lambda ,{\mathcal{Q}}_i\nu {\left(i\right)}^{\top }\rangle =\langle \lambda ,\sum _{i=1}^{\mathsf{\ell }}{\mathcal{Q}}_i\nu {\left(i\right)}^{\top }\rangle =\langle \lambda ,\mathcal{Q}{\nu }^{\top }\rangle =\langle \lambda ,L\left(\nu \right)\rangle =0.$$

   □

The following proposition can easily be verified by straightforward computations.

Proposition 2.

For $\nu \in {\mathbb{N}}_0^{\mathfrak{m}}$, $\nu \in {\mathcal{M}}_L$ if and only if the monomial $a^{\nu }$ is a first integral of the linear system

$$\dot{a}=\mathcal{L}a,$$

where

$$\mathcal{L}=\mathrm{diag}\left({\lambda }^{\top }\mathcal{Q}\right).$$

From Lemma 1 and Proposition 2, we conclude that ${\mathcal{M}}_L$ has a Hilbert basis $H_L$, which can be computed using Algorithm 1 above. This algorithm works in the general case where the ${\lambda }_i$ are algebraic elements of $\mathbb{C}$, and is not limited to the case where all ${\lambda }_i$ are rational numbers. In the latter simple case, we may use Algorithm 1.4.5 from [28].

Example 2.

Consider the following differential system, written in the form (20), with linear part as in Example 1,

$$\dot{x}=x\odot \lambda +x\odot a_{Q_1}{x}^{Q_1}+x\odot a_{Q_2}{x}^{Q_2}+x\odot a_{Q_3}{x}^{Q_3},$$

where $\lambda =\left(1,\zeta ,{\zeta }^2,-2,3\right)$ and $Q_1=(1,0,0,0,1),Q_2=(0,1,1,0,0)$, and $Q_3=(1,0,0,1,1)$. In particular, the complex vectors of coefficients are

$$\begin{array}{cc}\hfill a_{Q_1}& =(a_{10001},b_{10001},c_{10001},d_{10001},f_{10001}),\\ \hfill a_{Q_2}& =(a_{01100},b_{01100},c_{01100},d_{01100},f_{01100}),\\ \hfill a_{Q_3}& =(a_{10011},b_{10011},c_{10011},d_{10011},f_{10011}).\end{array}$$

Following the above notations, let $\mathcal{Q}=\left[{\mathcal{Q}}_1{\mathcal{Q}}_2{\mathcal{Q}}_3\right]$ be the $5\times 15$ matrix

$$\mathcal{Q}=\left[\begin{array}{ccccccccccccccc}1& 1& 1& 1& 1& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1\\ 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1\end{array}\right]$$

and

$${\lambda }^{\top }\mathcal{Q}=(4,4,4,4,4,-1,-1,-1,-1,-1,2,2,2,2,2).$$

To obtain the Hilbert basis for the monoid ${\mathcal{M}}_L$ of solutions $\nu \in {\mathbb{N}}_0^{15}$, we use Algorithm 1 with the input vector ${\lambda }^{\top }\mathcal{Q}$. It turned out that the basis has 425 vectors, which are the exponent vectors of the monomials returned by the algorithm. Since the list is rather long, we do not present it in the paper, but we present the computations for the particular case when

$$b_{10001}=d_{10001}=c_{01100}=d_{01100}=a_{10011}=b_{10011}=d_{10011}=0.$$

In this case, 211 binomials are generated by the algorithm. Only 64 binomials are of the form ${\mathbf{x}}^{\nu }-{\mathbf{y}}^{\nu }$. The leading terms of the binomials are given in Appendix C. Using them, we obtain the Hilbert basis of ${\mathcal{M}}_L$, which consists of the 64 vectors in ${\mathbb{N}}_0^{15}$. In Appendix D, we present a Macaulay2 code used to compute the Hilbert basis of ${\mathcal{M}}_L$. Using this Hilbert basis, we get the following invariants of group action (22) on our system:

$$\begin{array}{cc}& f_{01100}^2{f}_{10011},f_{01100}^2{c}_{10011},d_{01100},b_{01100}{f}_{01100}{f}_{10011},b_{01100}{f}_{01100}{c}_{10011},\\ & b_{01100}^2{f}_{10011},b_{01100}^2{c}_{10011},a_{01100}{f}_{01100}{f}_{10011},a_{01100}{f}_{01100}{c}_{10011},a_{01100}{b}_{01100}{f}_{10011},\\ & a_{01100}{b}_{01100}{c}_{10011},a_{01100}^2{f}_{10011},a_{01100}^2{c}_{10011},f_{10001}{f}_{01100}^4,f_{10001}{b}_{01100}{f}_{01100}^3,\\ & f_{10001}{b}_{01100}^2{f}_{01100}^2,f_{10001}{b}_{01100}^3{f}_{01100},f_{10001}{b}_{01100}^4,f_{10001}{a}_{01100}{f}_{01100}^3,\\ & f_{10001}{a}_{01100}{b}_{01100}{f}_{01100}^2,f_{10001}{a}_{01100}{b}_{01100}^2{f}_{01100},f_{10001}{a}_{01100}{b}_{01100}^3,\\ & f_{10001}{a}_{01100}^2{f}_{01100}^2,f_{10001}{a}_{01100}^2{b}_{01100}{f}_{01100},f_{10001}{a}_{01100}^2{b}_{01100}^2,f_{10001}{a}_{01100}^3{f}_{01100},\\ & f_{10001}{a}_{01100}^3{b}_{01100},f_{10001}{a}_{01100}^4,c_{10001}{f}_{01100}^4,c_{10001}{b}_{01100}{f}_{01100}^3,c_{10001}{b}_{01100}^2{f}_{01100}^2,\\ & c_{10001}{b}_{01100}^3{f}_{01100},c_{10001}{b}_{01100}^4,c_{10001}{a}_{01100}{f}_{01100}^3,c_{10001}{a}_{01100}{b}_{01100}{f}_{01100}^2,\\ & c_{10001}{a}_{01100}{b}_{01100}^2{f}_{01100},c_{10001}{a}_{01100}{b}_{01100}^3,c_{10001}{a}_{01100}^2{f}_{01100}^2,c_{10001}{a}_{01100}^2{b}_{01100}{f}_{01100},\\ & c_{10001}{a}_{01100}^2{b}_{01100}^2,c_{10001}{a}_{01100}^3{f}_{01100},c_{10001}{a}_{01100}^3{b}_{01100},c_{10001}{a}_{01100}^4,\\ & a_{10001}{f}_{01100}^4,a_{10001}{b}_{01100}{f}_{01100}^3,a_{10001}{b}_{01100}^2{f}_{01100}^2,a_{10001}{b}_{01100}^3{f}_{01100},a_{10001}{b}_{01100}^4,\\ & a_{10001}{a}_{01100}{f}_{01100}^3,a_{10001}{a}_{01100}{b}_{01100}{f}_{01100}^2,a_{10001}{a}_{01100}{b}_{01100}^2{f}_{01100},a_{10001}{a}_{01100}{b}_{01100}^3,\\ & a_{10001}{a}_{01100}^2{f}_{01100}^2,a_{10001}{a}_{01100}^2{b}_{01100}{f}_{01100},a_{10001}{a}_{01100}^2{b}_{01100}^2,\\ & a_{10001}{a}_{01100}^3{f}_{01100},a_{10001}{a}_{01100}^3{b}_{01100},a_{10001}{a}_{01100}^4.\end{array}$$

To conclude this section, we remark on the connection between the invariants studied here and those of classical invariant theory. Classical invariant theory is primarily concerned with polynomial functions that are invariant under the action of finite groups. In contrast, the present work considers the action of an infinite group on the phase space. This action induces a representation on the parameter space, and our analysis focuses on the polynomial invariants in these parameters. This framework for invariants traces back to the seminal contributions of Sibirsky and his school [12,13].

5. Normal Forms

In this section, we discuss the structure of $C_0^{for}\left(A\right)$ and $C^{pol}\left(A\right)$, as defined by Equation (7), and present an algorithm for computing ${\mathcal{N}}_{\lambda }^{\left(k\right)}$.

By [9,11], $C^{pol}$ is an $I\left(\lambda \right)$-module. We have an algorithm to describe $I\left(\lambda \right)$, so we need to find only a generating set of this $I\left(\lambda \right)$-module.

As before, let $\lambda ={({\lambda }_1,\dots ,{\lambda }_n)}^{\top }$, ${\mathbf{N}}_i=\{Q\in {\mathbb{Z}}^n:Q_i\ge -1,Q_j\ge 0\mathrm{if}j\ne i\}$ and $\mathbf{N}={\mathbf{N}}_{\mathbf{1}}\cup \cdots \cup {\mathbf{N}}_{\mathbf{n}}$. To find a generating set of equivariants, we need to describe the sets

$${\mathcal{N}}_{\lambda }^{\left(k\right)}=\{\alpha \in {\mathbb{N}}_0:\langle \alpha ,\lambda \rangle -{\lambda }_k=0\}k=1,\dots ,n.$$

Let

$$\widetilde{{\mathcal{N}}_{\lambda }^{\left(k\right)}}=\{\beta \in {\mathbf{N}}_k:\langle \beta ,\lambda \rangle =0\}k=1,\dots ,n.$$

The following statement is obvious.

Proposition 3.

(1) For any $k=1,\dots ,n$ the monoid ${\mathcal{M}}_{\lambda }$ is a subset of $\widetilde{{\mathcal{N}}_{\lambda }^{\left(k\right)}}.$

(2) There is one-to-one correspondence between the elements of ${\mathcal{N}}_{\lambda }^{\left(k\right)}$ and elements of $\widetilde{{\mathcal{N}}_{\lambda }^{\left(k\right)}}$.

To describe ${\mathcal{N}}_{\lambda }^{\left(k\right)}$, recall the $d\times n$ matrix $\mathfrak{A}$ from (13), which resulted from the equation ${\sum }_{i=1}^n{\alpha }_i{\lambda }_i=0$. To solve ${\sum }_{i=1}^n{\alpha }_i{\lambda }_i={\lambda }_k$, we introduce a slack variable ${\alpha }_0$ and solve the following homogeneous equation in $({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0^{n+1}$:

$$-{\alpha }_0{\lambda }_k+\sum _{i=1}^n{\alpha }_i{\lambda }_i=0.$$

(27)

This can be accomplished by applying Algorithm 1 to the following $d\times (n+1)$ matrix:

$${\mathfrak{A}}_k=[-{\mathfrak{a}}_k{\mathfrak{a}}_1{\mathfrak{a}}_2\cdots {\mathfrak{a}}_n],$$

(28)

where ${\mathfrak{a}}_j$ denotes the j-th column of $\mathfrak{A}$ (13).

In the resulting Hilbert basis, the set of vectors where ${\alpha }_0=0$ is precisely the Hilbert basis $H_{\lambda }$ of ${\mathcal{M}}_{\lambda }$, while those vectors where ${\alpha }_0=1$ are solutions to ${\sum }_{i=1}^n{\alpha }_i{\lambda }_i={\lambda }_k$. Let S be the set of all such vectors. It is clear that any solution to Equation (27) is of the form $\nu +\mu$ where $\nu \in S$ and $\mu \in {\mathcal{M}}_{\lambda }$.

Algorithm 3 is an extension of Algorithm 1, incorporating the procedure above and producing a minimal generating set for ${\mathcal{N}}_{\lambda }^{\left(k\right)}$. The correctness of this algorithm follows from the correctness of Algorithm 1 and the discussion above.

Algorithm 3: Minimal generating set of ${\mathcal{N}}_{\lambda }^{\left(k\right)}$

Input: A vector $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{C}}^n$ of algebraic elements Output: A minimal generating set for ${\mathcal{N}}_{\lambda }^{\left(k\right)}$ 1  Find a number field K containing all ${\lambda }_i$ 2  Choose a $\mathbb{Q}-$basis $\{b_1,\dots ,b_d\}$ for K 3  Express each ${\lambda }_i$ in the chosen basis: Write $${\lambda }_i=c_{1i}{b}_1+c_{2i}{b}_2+\cdots +c_{di}{b}_d,\mathrm{where} c_{ji}\in \mathbb{Q}$$ 4  Let $C=[-{\mathfrak{c}}_k{\mathfrak{c}}_1{\mathfrak{c}}_2\cdots {\mathfrak{c}}_n]$ be the $d\times (n+1)$-matrix of coefficients, where ${\mathfrak{c}}_i$ is the $d-$column vector ${\mathfrak{c}}_i=(c_{1i},c_{2i},\dots ,c_{di})$ 5  Convert to integer matrix: Let ${\mathfrak{A}}_k=[-{\mathfrak{a}}_k{\mathfrak{a}}_1{\mathfrak{a}}_2\cdots {\mathfrak{a}}_n]$ be the integer $d\times (n+1)$-matrix obtained from $\mathfrak{C}$ by clearing the denominators 6  Compute the reduced Gröbner basis $\mathcal{G}$ for the ideal $$〈{\displaystyle x_0-z_0\prod _{j=1}^d{t}_j^{-a_{jk}},x_i-z_i\prod _{j=1}^d{t}_j^{a_{ji}}:i=1,\dots ,n}〉$$ in $k[t_1,\dots ,t_d,t_1^{-1},\dots ,t_d^{-1},x_0,x_1,\dots ,x_n,z_0,z_1,\dots ,z_n]$ with respect to ≻: $$\{t_1^{\pm },\dots ,t_d^{\pm }\}\succ \{x_0,x_1,\dots ,x_n\}\succ \{z_0,z_1,\dots ,z_n\}.$$ 7  The Hilbert basis $H_{\lambda }$ of ${\mathcal{M}}_{\lambda }$ consists of all vectors $\mu$ such that $x_1^{{\mu }_1}\cdots x_n^{{\mu }_n}-z_1^{{\mu }_1}\cdots z_n^{{\mu }_n}$ appears in $\mathcal{G}$. 8  Let S be the set of all vector $\nu$ such that $x_0{x}_1^{{\nu }_1}\cdots x_n^{{\nu }_n}-z_0{z}_1^{{\nu }_1}\cdots z_n^{{\nu }_n}$ appears in $\mathcal{G}$. 9  Every vector in ${\mathcal{N}}_{\lambda }^{\left(k\right)}$ is of the form $\nu +\mu$ for some $\nu \in S$ and $\mu \in {\mathcal{M}}_{\lambda }$.

Example 3.

Consider system (16) from Example 1. Employing Algorithm 3 as shown in Appendix E, results in the following:

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{\lambda }^1& =& \{\nu +\mu :\nu \in \{(1,0,0,0,0),(0,0,0,1,1),(0,2,2,0,1)\}\mathit{and} \mu \in {\mathcal{M}}_{\lambda }\}\hfill \\ \hfill {\mathcal{N}}_{\lambda }^2& =& \{(0,1,0,0,0)+\mu :\mu \in {\mathcal{M}}_{\lambda }\}\hfill \\ \hfill {\mathcal{N}}_{\lambda }^3& =& \{(0,0,1,0,0)+\mu :\mu \in {\mathcal{M}}_{\lambda }\}\hfill \\ \hfill {\mathcal{N}}_{\lambda }^4& =& \{\nu +\mu :\nu \in \{(0,0,0,1,0),(0,2,2,0,0)\}\mathit{and} \mu \in {\mathcal{M}}_{\lambda }\}\hfill \\ \hfill {\mathcal{N}}_{\lambda }^5& =& \{\nu +\mu :\nu \in \{(0,0,0,0,1),(3,0,0,0,0)\}\mathit{and} \mu \in {\mathcal{M}}_{\lambda }\}\hfill \end{array}$$

In particular, in addition to the obvious equivariants of (16):

$$v_i=x_i{e}_{i}i=1,\dots ,5,$$

there are four additional equivariants:

$$v_6=x_4{x}_5{e}_1,v_7=x_2^2{x}_3^2{e}_4{v}_8=x_1^3{e}_5,v_9=x_2^2{x}_3^2{x}_5{e}_1.$$

Let $I_1,\dots ,I_6$ be the monomials corresponding to the elements of $H_{\lambda }$ which are given in (17). Then, every element of $C^{pol}\left(A\right)$ can be written as

$$\begin{array}{c}{x}_1{e}_1{f}_1(I_1,\dots ,I_6)+x_2{e}_2{f}_2(I_1,\dots ,I_6)+x_3{e}_3{f}_3(I_1,\dots ,I_6)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_4{e}_4{f}_4(I_1,\dots ,I_6)+x_5{e}_5{f}_5(I_1,\dots ,I_6)+x_4{x}_5{e}_1{f}_6(I_1,\dots ,I_6)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_2^2{x}_3^2{e}_4{f}_7(I_1,\dots ,I_6)+x_1^3{e}_5{f}_8(I_1,\dots ,I_6)+x_2^2{x}_3^2{x}_5{f}_9(I_1,\dots ,I_6),\hfill \end{array}$$

(29)

for some $f_1,\dots ,f_9\in \mathbb{C}[I_1,\dots ,I_6]$. However, it is easy to verify the following syzygies:

$$\begin{array}{ccc}\hfill I_4{v}_6-I_1{v}_1& =& 0,I_5{v}_6-I_2{v}_1=0,I_6{v}_6-I_4{v}_1=0\hfill \\ \hfill I_1{v}_7-I_2^2{v}_4& =& 0,I_2{v}_7-I_3{v}_4=0,I_4{v}_7-I_2{I}_5{v}_4=0,I_6{v}_7-I_5^2{v}_5=0\hfill \\ \hfill I_1{v}_8-I_4{I}_6{v}_5& =& 0I_2{v}_8-I_5{I}_6{v}_5=0,I_3{v}_8-I_6^2{v}_5=0\hfill \\ \hfill I_1{v}_9-I_2^2{v}_1& =& 0,I_2{v}_9-I_3{v}_6=0,I_4{v}_9-I_2^2{v}_1=0,I_5{v}_9-I_3{v}_1=0.\hfill \end{array}$$

Using these syzygies, we can write (29) in a unique way in the simpler form

$$\begin{array}{c}{f}_1(I_1,\dots ,I_6)v_1+f_2(I_1,\dots ,I_6)v_2+f_3(I_1,\dots ,I_6)v_3+f_4(I_1,\dots ,I_6)v_4+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{f}_5(I_1,\dots ,I_6)v_5+f_6(I_1,I_2,I_3)v_6+f_7(I_3,I_5)v_7+f_8(I_4,I_5,I_6)v_8+f_9(I_3,I_6)v_9.\hfill \end{array}$$

This leads to the following representation of $C^{pol}$:

$$\begin{array}{c}{C}^{pol}\left(A\right)=\mathbb{C}[I_1,\dots ,I_6]v_1\oplus \mathbb{C}[I_1,\dots ,I_6]v_2\oplus \mathbb{C}[I_1,\dots ,I_6]v_3\oplus \mathbb{C}[I_1,\dots ,I_6]v_4\oplus \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbb{C}[I_1,\dots ,I_6]v_5\oplus \mathbb{C}[I_1,I_2,I_3]v_6\oplus \mathbb{C}[I_3,I_5]v_7\oplus \mathbb{C}[I_4,I_5,I_6]v_8\oplus \mathbb{C}[I_3,I_6]v_9.\hfill \end{array}$$

By replacing the polynomial ring in the above formula with the ring of formal power series, we obtain a representation of $C_0^{for}$, which is called a Stanley decomposition of the normal form module [22].

Suppose that system (1) is in the normal form and system (8) admits p independent polynomial first integrals. By Proposition 5 of [6], system (1) admits p independent formal first integrals if and only if it admits every polynomial first integral of (8). The following statement provides a slight extension of this result.

Proposition 4.

Let $\varphi :{\mathbb{Z}}^n\to \mathbb{C}$ be defined by $\varphi \left(x\right)=\langle \lambda ,x\rangle .$ Suppose that

$$\mathrm{rank}ker\varphi =\mathrm{rank}{\mathcal{R}}_{\lambda }=p,$$

(30)

and system (1) is in the normal form. Then, system (1) admits p functionally independent formal first integrals if and only if it admits every monomial first integral of (8) in the ring of Laurent polynomials.

Proof. 

First we note that the Laurent monomials $x^{{\alpha }_1},\dots ,x^{{\alpha }_m}$ are functionally independent if and only if their exponents ${\alpha }_1,\dots ,{\alpha }_m$ are linearly independent over $\mathbb{C}$. Assume system (1), which is in the normal form, admits every Laurent monomial first integral of (8). Since $\mathrm{rank}{\mathcal{R}}_{\lambda }=p$, there are ${\alpha }_1,\dots ,{\alpha }_p\in {\mathbb{N}}_0^n$, which are linearly independent over $\mathbb{Z}$. Then, ${\alpha }_1,\dots ,{\alpha }_p$ are also linearly independent over $\mathbb{C}$. Therefore, $x^{{\alpha }_1},\dots ,x^{{\alpha }_p}$ are functionally independent monomial first integrals, which can also be considered as formal first integrals.

Suppose now that system (1), which is in the normal form, admits p formal first integrals. Then (8) admits p polynomial first integrals. By Proposition 5 of [6], Since $\mathrm{rank}{\mathcal{R}}_{\lambda }=p$, system (1) admits all monomial first integrals of (8). By (30), every solution $\alpha$ to (11) is a $\mathbb{Z}$-linear combination of elements of ${\mathcal{H}}_{\lambda }$, yielding that the corresponding $x^{\alpha }$ is a Laurent monomial first integral of (1) (which is in the normal form).    □

In general, checking if (30) holds can be a difficult problem. However, in the case where all eigenvalues of A are algebraic elements over $\mathbb{Q}$, both ${\mathcal{R}}_{\lambda }$ and its rank can be computed using Algorithm 2. Instead of computing the rank of $ker\varphi$, we can easily compute the rank of $K_{\rho }$ defined by (14). In particular, for system (16) of Example 1, we have $\mathrm{rank}{K}_{\rho }=\mathrm{rank}{\mathcal{R}}_{\lambda }=3$. Therefore, by Proposition 4, if a normal form of an analytic or formal system with the linear part (16) admits 3 independent first integrals, then it admits every Laurent monomial first integral of system (16).

6. Conclusions

In this work, we presented an algorithm for computing the generating set of the algebra of polynomial first integrals of system (8). We further adapted this algorithm to compute the generating set of polynomial invariants of system (18) under action (21). This approach was also applied to analyze the structure of the Poincaré-Dulac normal form of system (1). An interesting and important direction for future research is to investigate the connection between ${\mathcal{N}}_{\lambda }^{\left(k\right)}$ and the Diophantine hull of A [10].