Reversible-equivariant systems and matricial equations

Abstracts

This paper uses tools in group theory and symbolic computing to classify the representations of finite groups with order lower than, or equal to 9 that can be derived from the study of local reversible-equivariant vector fields in <img border=0 width=32 height=32 src="../../../../img/revistas/aabc/v83n2/carr.jpg" align=absmiddle>4 . The results are obtained by solving matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form.

Reversible-equivariant dynamical systems; involutory symmetries; normal forms

Este artigo utiliza ferramentas da teoria de grupos e computação simbólica para dar uma classificação das representações de grupos finitos de ordem menor ou igual a 9 que podem ser consideradas no estudo local de campos vetoriais reversíveis-equivariantes em <img border=0 width=32 height=32 src="../../../../img/revistas/aabc/v83n2/carr.jpg" align=absmiddle>4 . Os resultados são obtidos resolvendo algebricamente equações matriciais. Em particular, exibimos as involuções utilizadas no estudo local de campos vetoriais reversíveis-equivariantes. Baseado em tal abordagem, nós apresentamos, para cada elemento desta classe, uma forma normal de Belitskii simplificada.

Sistemas dinâmicos reversíveis-equivariantes; simetrias involutórias; formas normais

MATHEMATICAL SCIENCES

Reversible-equivariant systems and matricial equations

Marco A. Teixeira; Ricardo M. Martins

Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas/UNICAMP, Rua Sérgio Buarque de Holanda, 651 Cidade Universitária, 13083-859 Campinas, SP, Brasil

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ABSTRACT

This paper uses tools in group theory and symbolic computing to classify the representations of finite groups with order lower than, or equal to 9 that can be derived from the study of local reversible-equivariant vector fields in

⁴ . The results are obtained by solving matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form.

Key words: Reversible-equivariant dynamical systems, involutory symmetries, normal forms.

RESUMO

Este artigo utiliza ferramentas da teoria de grupos e computação simbólica para dar uma classificação das representações de grupos finitos de ordem menor ou igual a 9 que podem ser consideradas no estudo local de campos vetoriais reversíveis-equivariantes em

⁴ . Os resultados são obtidos resolvendo algebricamente equações matriciais. Em particular, exibimos as involuções utilizadas no estudo local de campos vetoriais reversíveis-equivariantes. Baseado em tal abordagem, nós apresentamos, para cada elemento desta classe, uma forma normal de Belitskii simplificada.

Palavras-chave: Sistemas dinâmicos reversíveis-equivariantes, simetrias involutórias, formas normais.

1 INTRODUCTION

The presence of involutory symmetries and involutory reversing symmetries is very common in physical systems, for example, in classical mechanics, quantum mechanics and thermodynamic (see Lamb and Roberts 1996). The theory of ordinary differential equations with symmetry dates back from 1915 with the work of Birkhoff. Birkhoff realized a special property of his model: the existence of a involutive map R such that the system was symmetric with respect to the set of fixed points of R. Since then, the work on differential equations with symmetries stay restricted to hamiltonian equations. Only in 1976, Devaney developed a theory for reversible dynamical systems.

In this paper, involutory symmetries and involutory reversing symmetries are considered with in aunified approach. We study some possible linearizations for symmetries and reversing symmetries, around a fixed point, and employ this to simplify normal forms for a class of vector fields.

In particular, using tools from group theory and symbolic computing, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form. This new normal form simplifies the study of qualitative dynamics, unfoldings, and bifurcations, as we have to deal with a smaller number of parameters.

An important point to mention is that any map possessing an involutory reversing symmetry is the composition of two involutions, as was found by Birkhoff in 1915. It is worth pointing out that properties of reversing symmetry groups are a powerful tool to study local bifurcation theory in presence of symmetries, see for instance Knus et al. (1998).

The authors are grateful to the referees for many helpful comments and suggestions.

2 STATEMENT OF MAIN RESULTS

Let

₀ (⁴) denote the set of all germs of C^∞ vector fields in ⁴ with an isolated singularity at origin.

Define

The condition αβ ≠ 0 is to keep the problem nondegenerate, while the condition α ≠ β is necessary to avoid the appearence of 1-parameter families of symmetries (when working with reversible-equivariant vector field).

Given a group G generated by involutive diffeomorphisms φ: ⁴ , 0 → ⁴ , 0 (involutive means ²= Id) and a group homomorphism ρ : G → {-1, 1}, we say that X ∈ ₀ ( ⁴ ) is G-reversible-equivariant if, for each φ ∈ G,

Dφ (x) X (x) = ρ (φ) X (φ (x)).

If K ⊆ G is such that ρ (K ) = 1 we say that X is K -equivariant. If K ⊂ G is such that ρ (K ) =-1, we say that X is K -reversible. It is clear that if X is φ₁-reversible and φ₂-reversible, then X is also φ₁φ₂-equivariant. It is usual to denote G₊={φ ∈ G; ρ (φ) = 1} and G-={φ ∈ G; ρ (φ) =-1}. Note that G₊ is a subgroup of G, but G_- is not.

If X ∈ ₀ ( ⁴ ) is φ-reversible (resp. φ-equivariant) and γ (t) is a solution of

with γ (0) = x₀, then φγ (-t) (resp. φγ (t)) is also a solution for (2). In particular, if X is a φ-reversible (or φ-equivariant) vector field, then the phase portrait of X is symmetric with respect to the subspace Fix (φ), in the sense that φ maps the phase portrait of X to itself, reversing the direction of time in the reversible case.

A survey on reversible-equivariant vector fields is described in Antoneli et al. (2009), Lamb and Roberts (1996), Devaney (1976) and references therein.

In this paper, we shall restrict our study to G-reversible-equivariant vector fields where G is finite, generated by two involutions {φ, ψ} and the group homomorphism ρ : G → {-1, 1} is given by

ρ (φ) = ρ (ψ) =-1. In this case, by a basic group theory argument, one can prove that there is n > 2 such that G

_n.

Our aim is to provide an analogous form of the theorem below to the G-reversible-equivariant case:

THEOREM 1. Let X ∈ ₀ (²ⁿ ) be a φ-reversible vector field, where φ :²ⁿ , 0 →²ⁿ , 0 is a C^∞ involution with dim Fix (φ) = n as a submanifold, and let R₀ :²ⁿ →²ⁿ be any linear involution with dim Fix (R₀) = n. Then there exists a C^∞ change of coordinates h :²ⁿ , 0 →²ⁿ , 0 (depending on R₀) such that h_* X is R₀-reversible.

The proof of Theorem 1 is straightforward: φ is locally conjugated to Dφ (0) by the change of coordinates Id +dφ (0) φ; now, Dφ (0) and R₀ are linearly conjugated (by P, say), as they are linear involutions with dim Fix (Dφ (0)) = dim Fix (R₀). Now take h =P(Id +dφ (0) φ).

Theorem 1 is very useful when one works locally with reversible vector fields. See for example in Buzzi et al. (2009) and Teixeira (1997). It allows to always fix the involution as the following:

DEFINITION 2. Given a finitely generated group G = (g₁, . . . , g_l) with the generations fixed, a representation σ : G → M_n×n () and a vector field X ∈ ₀ (ⁿ), we say that the representation σ is (X, G) compatible if σ (g_j) X (x) =-X (σ (g_j)), for all j = 1, . . . , l.

We prove the following:

THEOREM A: Given X ∈ (

⁴), we present all the (X, n) -compatible 4-dimensional representations of X, for n = 2, 3, 4.

As an application of Theorem A, we obtain the following result.

THEOREM B: The Belitskii normal form for

₄-reversible-equivariant vector fields in (

⁴ ) is exhibited, for α, β odd integers with (α, β) = 1.

For further details on normal form theory, see Belitskii (2002) and Bruno (1989).

This paper is organized as follows. In Section 3 we set the problem and reduce it to a system of matricial equations. In Section 4 we prove Theorem A and in Section 5, we prove Theorem B.

3 SETTING THE PROBLEM

Consider X ∈ (

⁴) for αβ ≠ 0. Denote A =DX (0). Let φ, ψ : ⁴ , 0 → ⁴ be involutions with dim Fix (φ) = dim Fix (ψ) = 2 and suppose that X is (φ, ψ) -reversible-equivariant.

Next result will be useful in the sequel.

THEOREM 3 (Bochner and Montgomery 1946). Let G be a compact group of C^k diffeomorphisms defined on a C^k>1 manifold

. Suppose that all diffeomorphisms in G have a common fixed point, say x₀. Then, there exists a C^k coordinate system h around x0 such that all diffeomorphisms in G are linear with respect to h.

Putting G = (φ, ψ), as φ (0) = 0 and ψ (0) = 0, Theorem 3 says that there exists a coordinate system h around 0 such that = h^-1φh and = h^-1ψ h are linear involutions. Now consider as X in this new system of coordinates, that is, = h_* X. Then is -reversible-equivariant.

Now choose any linear involution R₀: ⁴ → ⁴ with dim Fix (R₀) =2. As R₀ and φ are linearly conjugated, we can pass to a new system of coordinates such that is -reversible-equivariant for some ₀. However, it is not possible to choose a priori a good (linear) representative for the second involution, ₀.

In other words, it is not possible to produce an analog version of Theorem 1 for reversible-equivariant vector fields. We shall take into account all the possible choices for the second involution.

PROBLEM A: Let G = φ, ψ be a group generated by involutive diffeomorphisms, and X ∈ ₀ (²ⁿ) be a G-reversible-equivariant vector field. Find all of the (X, G) -compatible representations σ with σ (φ) = R₀, R₀ given by (3).

To solve Problem A, we have to determine all the linear involutions S such that (R₀, S) G and SDX (0) + DX (0) S = 0 (this last relation is the compatibility condition for the linear part of X).

4 PROOF OF THEOREM A

In this section we prove Theorem A, that deals with (X, _n) -compatible representations for n = 2, 3, 4, that is, the list of groups to be considered is: ₂= (

₂ × ₂ ), ₃ and ₄ .

In the rest of this section we denote by A the matrix A (α, β) defined in (1).

4. 1 CASE

₂ × ₂

Fix the matrix

Note that R₀² = Id and R₀A =-AR₀. We need to determine all possible involutive matrices S ∈ ^4×4 such that

SA =-AS

and

(R₀, S) =

₂ × ₂ .

Note that the relation R₀, S

₂ × ₂ is equivalent to R₀S =SR₀ and S² Id.

Put

The relations SA =-AS, S² = Id and R₀S =SR₀ are represented by the following systems ofpolynomial equations:

LEMMA 4. System (6) has 4 solutions:

PROOF. This can be done in Maple 12 by means of the Reducefunction from the Groebner package and the usual Maple's solvefunction. We remark that the solution S4 is degenerate, i. e. , S4 = R₀. Moreover, we remark that the above representations of ₂ × ₂ are not equivalent.

Now we state the main result for

₂ × ₂ -reversible vector fields. With the notation of Section 3, it assures that the linear involutions S_jare the unique possibilities for ₀.

THEOREM 5. Let Ω⊂ (

⁴) be the set of

₂×₂-reversible-equivariant vector fields X ∈ (

⁴ ). Then Ω= Ω₁∪Ω₂∪Ω₃, where X ∈ Ω_j if X is (R₀, S_j) -reversible-equivariant in some coordinate system around the origin.

PROOF. Let X ∈ Ω. Then there are distinct and nontrivial involutions φ, ψ : ⁴ , 0 → ⁴ , 0 with φψ = ψφ such that X is φ-reversible and ψ-reversible. By Theorem 3, there is a system of coordinates around the origin where X is R₀ -reversible and S0-reversible, with R₀ (x₁, x₂, y₁, y₂) = (x₁, -x₂, y₁, -y₂) and S₀ a linear involution with R₀S₀ = S₀R₀. By Lemma 4 S₀ ∈{S₁, S₂, S₃, S₄}. As φ = ψ, S₀ = S₄. So X is R₀ -reversible and S_j-reversible for some j ∈{1, 2, 3}. Then X ∈ Ω₁∪Ω₂∪Ω₃.

REMARK 6. Theorem 5 can be proved without using Lemma 4. The following technique, that also applies to Theorems 11 and 14, was communicated to us by one of the referees.

Consider the decomposition of

⁴ in the generalized eigenspaces ⁴ = Aut (α) ⊕ Aut (β), where

Then the equation SA =-AS keeps the above decomposition fixed (for α ≠ β). This reduces the problem of determining S to two two-dimensional linear problems, and can be easily generalized to arbitrary dimensions. We keept the algorithmic proofs just because we are more familiar with the computational approach.

Now let us give a characterization of the vector fields which are (R₀, S_j) -reversible. Let us fix

with x ≡ (x₁, x₂, y₁, y₂). The proof of next results will be omitted.

COROLLARY 7. The vector field (7) is (R₀, S₁) -reversible if and only if the functions f j satisfy

In particular, f_{1, 3} (x₁, 0, y₁, 0) ≡ 0 and f_{2, 4} (0, x₂, 0, y₂) ≡ 0.

COROLLARY 8. The vector field (7) is (R₀, S₂) -reversible if and only if the functions f j satisfy

In particular, f_{1, 3} (x₁, 0, y₁, 0) ≡ 0 and f_{2, 3} (0, x₂, y₁, 0) ≡ 0.

COROLLARY 9. The vector field (7) is (R₀, S₃) -reversible if and only if the functions f j satisfy

In particular, f_{1, 3} (x₁, 0, y₁, 0) ≡ 0 and f_1,4 ( x₁, 0,0, y₂, 0) ≡ 0.

4. 2 CASE

₃

As above we fix the matrix

Now we need to determine all possible involutive matrices S ∈ R^4×4 such that

SA =-AS

and

(R₀, S) =

₃ .

Considering again

the equations SA + AS = 0, S² - Id = 0 and (R₀S) ³ - Id = 0 are equivalent to a huge system of equations. Their expression will be not presented.

LEMMA 10. The system generated by the above conditions has the following non degenerate solutions:

PROOF. Again, the proof can be done in Maple 12 using the Reducefunction from the Groebner package and the usual Maple's solve function.

At this point, we can state the following:

THEOREM 11. Let Ω₃ ⊂ (

⁴) be the set of

₃-reversible-equivariant vector fields X ∈ (

⁴). Then Ω₃ = Ω₃ ∪ Ω₂ ∪ Ω₃, where X ∈ Ω_j if X is (R₀, S_j ) -reversible-equivariant in some coordinate system around the origin.

PROOF. This proof is very similar to that of Theorem 5.

Now we present some results in the sense of Corollary 7 applied to

₃ -reversible vector fields. The characterization is given just for the (R₀, S₂) -reversible vector fields. Similar statements for (R₀, S_j) reversible vector fields, with j ∈{1, 3} can be easily deduced.

Let us fix again

with x ≡ (x₁, x₂, y₁, y₂). Keeping the same notation of Section 4. 1, we have now (R₀, S_j

₃ . We consider for instance j = 2.

COROLLARY 12. The vector field (12) is (R₀, S₂) -reversible if and only if the functions g j satisfy

And

In particular g_{1, 3} (x₁, 0, y₁, 0) ≡ 0 and g_{2, 4} (0, x₂, 0, y₂) ≡ 0.

The next section deals with the characterization of the

₄ -reversible vector fields. The analysis of the ₃ -reversible case will be omitted since it is very similar to the ₄ -reversible case and this last case is more interesting (there are more representations).

4. 3 CASE

₄

Fix the matrix

Again, our aim is to determine all the possible involutive matrices S ∈ ^4×4 such that

SA =-AS

and

(R₀, S)

₄ .

Considering again

the equations SA + AS = 0, S² - Id = 0 and (R₀S)⁴ - Id = 0 are represented by a easily deduced buthuge system having 12 non degenerate solutions, arranged in the following way:

Recall that the above arrangement has obeyed the rule:

LEMMA 13. S_i , S_j∈ _k ⇔ (R₀, Si ) = (R₀, S_j).

For each i ∈{1, . . . , 6}, denote by Si one of the elements of _i. The proof of the next result follows immediately the above lemmas.

THEOREM 14. Let Ω₄ ⊂(

⁴) be the set of

₄-reversible-equivariant vector fields X ∈(⁴). Then Ω₄ = Ω₁ ∪ Ω₂ ∪...∪ Ω₆, where X ∈ Ω_j if X is (R₀, S_j) -reversible-equivariantin some coordinate system around the origin.

PROOF. This proof is very similar to that of Theorem 5. It will be omitted.

Now we present some results in the sense of Corollary 7 applied to

₄ -reversible vector fields. Thecharacterization is given just for the (R₀, S₁) -reversible vector fields. Similar statements for (R₀, S_j) reversible vector fields, with j ∈{2, 3, 4, 5, 6} can be easily deduced.

Let us fix again

with x ≡ (x₁, x₂, y₁, y₂). Keeping the same notation of Section 4. 1, we have now (R₀, S_j

₄ .

COROLLARY 15. The vector field (15) is (R₀, S₁) -reversible if and only if the functions g j satisfy

In particular g_{1, 3} (x₁, 0, y₁, 0) ≡ 0 and g_{2, 4} (0, x₂, 0, y₂) ≡ 0.

5 APPLICATIONS TO NORMAL FORMS (PROOF OF THEOREM B)

Let X ∈(

⁴) be a ₄-reversible vector field and X^N its reversible-equivariant Belitskii normal form.

To compute the expression of X^N , we have to consider the following possibilities of the parameter λ= αβ^-1:

(i) λ

,

(ii) λ = 1,

(iii) λ = pq^-1 ≠ 1, with p, q integers and ( p, q) = 1.

In the case (i), one can show that the normal forms for the reversible and reversible-equivariant cases are essentially the same. This means that any reversible field with such linear approximation is automatically reversible-equivariant. In view of this, case (i) is not interesting, and its analysis will be omitted. We just note that case (ii) will not be discussed here because of its deep degeneracy, as the range of its homological operator

is a very low dimensional subspace of H^k. Also recall that we suppose α ≠ β in the definition of A (α, β).

Our goal is to focus on the case (iii). Put α = p and β = q, with p, q ∈ and ( p, q) = 1. How to compute a normal form which applies for all ₄ -reversible vector fields, without choosing specific involutions?

According to the results in the last section, it suffices to show that X^N satisfies

and

with S_jgiven on Lemma 13, as the fixed choice for the representative of _i .

First of all, we consider complex coordinates (z₁, z₂) ∈ ² instead of (x₁, x₂, y₁, y₂) ∈ ⁴:

We will write (z) for the real part of the complex number z and (z) for its imaginary part.

Define

Note that each Δj corresponds to a resonance relation among the eigenvalues of the matrix A ( p, q) (given in (1)). For instance, if λ₁ = pi and λ₂ = qi, then

for all m, n > 1. Then

is a resonance relation, and this relation corresponds to the resonant monomial

Collecting all the resonant monomials, the complex Belitskii normal form for X in this case is expressedby

with f_j , g_jwithout linear and constant terms. For more details on the construction of this Belitskii normal form, we refer to Belitskii (2002).

Now we consider the effects of

₄ -reversibility on the system (18). Writing our involutions in complex coordinates, we derive immediately that

LEMMA 16. Let

Then each group (φ₀, φ_j) corresponds to (R₀, S_j), j = 1, . . . , 6.

To compute a

₄ -reversible normal form for a vector field, one has first to define which of the groups in Lemma 16 can be used to do the calculations. Now we establish a normal form of a ₄ -reversible and p: q-resonant vector field X, depending only on p, q and not on the involutions generating ₄:

THEOREM 17. Let p, q be odd numbers with pq > 1 and X ∈ (

⁴ ) be a D4-reversible vector field.

Then X is formally conjugated to the following system:

with a_{i j} , b_{i j}∈ depending on j^k X (0), for k =i +j.

REMARK 18. The hypothesis on p, q given in Theorem 17 can be relaxed. In fact, if p, q satisfies the following conditions

then the conclusions of Theorem 17 are also valid (see Martins 2008).

REMARK 19. The normal form (19) coincides (in the nonlinear terms) with the normal form of a reversible vector field X ∈ (

⁴) with αβ^-1

. Remember that this fact allowed us to discard the case αβ^-1

at the beginning of this section.

The proof of Theorem 17 (even with the hypothesis of Remark 18) is based on a sequence of lemmas. The idea is just to show that with some hypothesis on p and q, all the coefficients of Δ₃ and Δ₄ in the reversible-equivariant analogue of (18) must be zero.

First let us focus on the monomials that are never killed by the reversible-equivariant structure.

LEMMA 20. Let ν = , a ∈ . So, for any j ∈{1, . . . , 6}, the φ_j-reversibility implies =-a (or (a) = 0). In particular, these terms are always present (generically) in the normal form.

PROOF. From

and

follows that =a.

Now let us see what happens with the monomials of type

. We mention that only for such monomials a complete proof will be presented. The other cases are similar. Moreover, we will give the statement and the proof as stated in Remark 18.

LEMMA 21. Let ν=, b ∈ . So, we establish the following tables:

PROOF. Let us give the proof for φ₂-reversibility. The proof of any other case is similar. Note that

Then, from φ₂ (v (z)) =-v (φ₂ (z)) we have = (-1) ^q-1i ^pb. Now we apply the hypotheses on p, q and the proof follows in a straightforward way.

Next corollary is the first of a sequence of results establishing that some monomial do not appear in the normal form:

COROLLARY 22. Let X ∈ (

⁴) be a (φ₀, φ_j) -reversible vector field. Then if

• q ≡ 1 mod 4 or

•q ≡ 3 mod 4 or

•q ≡ 0 mod 4 and p +q odd or

•q ≡ 2 mod 4 and p +q even,

then the normal form of X does not contain monomials of the form

PROOF. Observe that the φ₀, φ_j-reversibility implies that the coefficients in (20) satisfy

REMARK 23. Note that if p, q are odd with pq > 1, then they satisfy the hypothesis of Corollary 22.

The following results can be proved in a similar way as we have done in Lemma 21 and Corollary 22.

PROPOSITION 24. Let X ∈ (

⁴ ) be a (φ₀, φ_j) -reversible vector field. If one of the following conditions is satisfied:

(i) q ≡ 1 mod 4,

(ii) q ≡ 3 mod 4,

(iii) q ≡ 0 mod 4 and p +q

(iv) q ≡ 2 mod 4 and p +q even,

then the normal form of X, given in (18), does not have monomials of type

PROPOSITION 25. Let X ∈ (

⁴) be a (φ₀, φ_j) -reversible vector field. If one of the following conditions is satisfied

(i) q ≡ 1 mod 4,

(ii) q ≡ 3 mod 4,

(iii) q ≡ 0 mod 4 and p +q odd,

(iv) q ≡ 2 mod 4 and p +q even,

then the normal form of X, given in (18), does not have monomials of type

REMARK 26. In fact, the conditions imposed on λ in the last results are used just to assure the (φ₀, φ_j) reversibility of the vector field X with j = 1. For 2 < j < 6, the normal form only contains monomials of type

Now, to prove Theorem 17, we have just to combine all lemmas, corollaries and propositions givenabove.

PROOF OF THEOREM 17. Note that the conditions imposed on λ in Theorem 17 fit into the hypothesis of Corollary 22 and Propositions 24 and 25. So, if p, q are odd numbers with pq > 1, then the normal form just have monomials of type

6 CONCLUSIONS

In Theorem A we have classified all involutions that make a vector field X ∈(

⁴) (φ, ψ) -reversible when the order of the group (φ, ψ) is smaller than 9.

We used the classification obtained in Theorem A to simplify the Belitskii normal form of

₄ -reversible vector fields in ⁴, according to their resonances.

A normal form can be used to study stability questions and also reveal hidden symmetries. It also paves the way to get first integrals and sometimes to show that the system is an integrable hamiltonian system. Moreover, the truncated normal form gives a good approximation (or at least an asymptotic one) for the original vector field. So it is very important to write the normal form as simply as possible.

Our results show that in some cases it is possible to write the normal form of

₄ -reversible vector fields near a resonant singularity in the simplest possible way, that is, without the resonant terms coming from the nontrivial resonant relation among the eigenvalues. This allows us to write, for instance, the center manifold reduction in the simplest possible way.

We remark that the same approach can be made to the discrete version of the problem, or when the singularity is not elliptic (see for example Jacquemard and Teixeira 2002).

One can easily generalize the results presented here mainly in two directions: for vector fields on higher dimensional spaces and for groups with higher order. In both cases the hard missions are to face the normal form calculations and to solve some very complicated system of algebraic equations.

ACKNOWLEDGMENTS

This research was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Brazil process No. 134619/2006-4 (Martins) and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Brazil projects numbers 2007/05215-4 (Martins) and 2007/06896-5 (Teixeira). Theauthors want to thank the referees for many helpful comments and suggestions.

Manuscript received on November 5, 2009; accepted for publication on August 13, 2010

AMS Classification: 34C20, 37C80, 15A24.

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Publication Dates

• Publication in this collection
  03 June 2011
• Date of issue
  June 2011

History

• Received
  05 Nov 2009
• Accepted
  13 Aug 2010