Journal of Lie Theory Volume 13 (2003) 279–289 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Local Integrating Factors

Dedicated to D. Pumplün on the occasion of his seventieth birthday

Sebastian Walcher

Communicated by P. Olver

Abstract. This is an investigation of local (analytic or formal) integrating factors near certain degenerate stationary points of plane analytic vector fields. The method is to use blow-ups and then apply known results from the non-degenerate case. The main result is that in general there is no formal integrating factor which is algebraic over the quotient field of the formal power series ring.

1. Introduction and preliminaries

Nonlinear symmetries of vector fields are hard to approach and to investigate, but they are relevant both for the structural and the qualitative understanding of ordinary differential equations. In dimension two, the situation is a little easier to understand, due to the correspondence between infinitesimal symmetries and integrating factors. The purpose of this article is to discuss integrating factors of analytic vector fields near degenerate stationary points, continuing the work in [13] and [14]. We note that another motivation for the investigation of integrating factors comes from differential algebra; see Prelle and Singer [10].

We will investigate degenerate stationary points using blow-up techniques. Thus we obtain detailed results on the possible structure of integrating factors in various cases, and as a consequence we find that in general there are not even formal integrating factors.

Let us introduce some notation and review known facts. (References to be consulted here are Lie [7], Olver [8, 9], Prelle and Singer [10], and [13], among others.) All functions and vector fields under consideration will be analytic, unless specified otherwise.

To a given differential equation ẋ = f(x) on an open, nonempty subset U of C^{2} , one associates the derivation L_{f} which assigns to a function ϕ : U → C its Lie derivative L_{f}(ϕ), with L_{f}(ϕ)(x) := Dϕ(x) f(x). A function ψ is called a

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag first integral, resp. a semi-invariant, of ẋ = f(x) (or, briefly, of f) if L_{f}(ψ) = 0, resp. L_{f}(ψ) = μψ for some analytic function μ on U. The set of zeros of a semi-invariant, as well as any level set of a first integral, is an invariant set for ẋ = f(x).

We call the vector field f divergence-free if div(f) := tr Df = 0. (This definition depends on the choice of coordinates.) To a scalar-valued function ϕ one assigns its Hamiltonian vector field q_{ϕ} := ({−∂ϕ/∂x_{2} \atop ∂ϕ/∂x_{1}} ), which is obviously divergence-free, and has the first integral ϕ by construction. Locally, a twodimensional divergence-free vector field is equal to the Hamiltonian vector field of some function, whence finding a first integral of a divergence-free vector field reduces to a quadrature problem. One directly verifies that div(ψf) = L_{f}(ψ) + div(f) · ψ, so ψf will be divergence-free if and only if L_{f}(ψ) + div(f) · ψ = 0. If ψ is nonzero, and this equation is satisfied, then we call ψ an integrating factor of ẋ = f(x). Another function ψ̃ is an integrating factor if and only if ψ̃/ψ is a first integral or constant. (It should be noted here that there are other versions of the differential equation in question, e.g. f_{2} dx_{1} − f_{1} dx_{2} = 0, which correspond to other, equivalent, notions of integrating factor.)

There is a well-known correspondence between integrating factors and infinitesimal orbital symmetries in dimension two: If g is a vector field such that [g, f] = λf for some analytic λ (with the Lie bracket defined by [g, f] (x) := Df(x) · g(x) − Dg(x) · f(x), as usual), then either θ(x) := det(f(x), g(x)) is identically zero, or θ^{−1} is an integrating factor for f. Conversely, one can construct infinitesimal orbital symmetries from integrating factors, cf. [13], for instance.

By C[[x_{1}, x_{2}]]_{c} (resp. C[[x_{1}, x_{2}]]) we denote the algebra of power series with a nontrivial region of convergence (resp. the algebra of formal power series); see Ruiz [11] for an account of the algebraic properties of these rings. We recall that both are noetherian and unique factorization domains, and that the invertible elements are precisely those with a nonzero constant term. Moreover, by C((x_{1}, x_{2}))_{c}, resp. C((x_{1}, x_{2})) we denote the respective quotient fields. We are interested in integrating factors of local analytic (or formal) vector fields that are algebraic over C((x_{1}, x_{2}))_{c}, resp. over C((x_{1}, x_{2})). As for one motivation, we recall the following classical result of Prelle and Singer ([10], Props. 1 and 2, specialized to our context):

Theorem1.1. Let ẋ = f(x) be given, with f an analytic or formal vector field.

(a) If this equation has a first integral that is elementary (in the sense of differential algebra) over C((x_{1}, x_{2}))_{c} (resp. C((x_{1}, x_{2}))) then it admits an integrating factor that is algebraic over C((x_{1}, x_{2}))_{c} (resp. C((x_{1}, x_{2}))).

(b) If the equation admits an integrating factor that is algebraic over C((x_{1}, x_{2}))_{c} (resp. C((x_{1}, x_{2}))) then it also admits an integrating factor of the specific form

(φ_{1}^{d_{1}} ⋯ φ_{r}^{d_{r}})^{−1},

with the φ_{i} non-invertible and irreducible semi-invariants, and rational numbers d_{i}. (The possibility r = 0 is included, with integrating factor 1.)

In the local theory it is sensible to reserve the notion “semi-invariant” for non-invertible series, and we will do so in the following. We will sometimes employ coordinate transformations in order to simplify a problem. Then an integrating factor ϕ_{1}^{d_{1}} ⋯ ϕ_{r}^{d_{r}} exp(μ) will be changed to the integrating factor (ϕ_{1} ∘ Ψ)^{d_{1}} ⋯ (ϕ_{r} ∘ Ψ)^{d_{r}} exp(ν) by a coordinate transformation Ψ, with exp(ν) also incorporating the functional determinant of Ψ; see [14], for instance.

2. Degenerate stationary points

We assume that f(0) = 0, thus f = ∑_{i≥1} f^{(i)} , with each f^{(i)} a homogeneous polynomial of degree i, and abbreviate B := f^{(1)} = Df(0). Let B = B_{s} +B_{n} be the decomposition into semisimple and nilpotent part, and α_{1} , α_{2} the eigenvalues of B. We call the stationary point 0 of the differential equation nondegenerate if B_{s} ≠ 0 (equivalently, not both α_{i} are zero), and degenerate otherwise. For nondegenerate stationary points we have the Poincaré-Dulac normal form at our disposal (see Bibikov [3], Bruno [4], and [12]): There is an invertible formal power series Ψ = id + … that is solution-preserving from ẋ = f(x) to ẋ = f̃(x), with f̃ = B + … a formal power series vector field in normal form, thus satisfying [B_{s}, f̃] = 0. For easy reference we quote some parts of a result recorded in [14], Thm. 2.3.

Lemma2.1. Let B_{s} = diag(α_{1}, α_{2}), with α_{1} ≠ 0. Let f be a (formal power series) vector field in normal form.

(a) If α_{2}/α_{1} is not a rational number then x_{1} and x_{2} are, up to multiplication with invertible series, the only irreducible semi-invariants of f = B_{s} in C[[x_{1}, x_{2}]]. There is no first integral that is algebraic over C((x_{1}, x_{2})), and the only integrating factor algebraic over C((x_{1}, x_{2})), up to multiplication by scalars, is (x_{1}x_{2})^{−1} .

(b) Let α_{2}/α_{1} = −q/p, with p and q relatively prime positive integers, p ≤ q, and f = B + ∑_{j≥1} γ^{j}(σ_{j}id + τ_{j}B), where σ_{j}, τ_{j} ∈ C, and γ(x) := x_{1}^{q}x_{2}^{p}. Then x_{1} and x_{2} are, up to multiplication with invertible series, the only irreducible semi-invariants of f in C[[x_{1}, x_{2}]]. If not all the σ_{j} are equal to zero then there exists no first integral that is algebraic over C((x_{1}, x_{2})). Up to multiplication by constants, the only integrating factor that is algebraic over C((x_{1}, x_{2})) is

(x_{1}^{1+lq}x_{2}^{1+lp}ρ(x))^{−1} = (x_{1}x_{2}γ(x)^{l}ρ(x))^{−1},

with l the smallest index such that σ_{l} ≠ 0, and

x_{1}x_{2}ρ(x) = det(B(x), σ_{l}x + ∑ γ(x)^{i}σ_{l+i}(x)).

(c) Let α_{2} = 0, and f = B + ∑_{j≥1} γ^{j}(σ_{j}id + τ_{j}B), where σ_{j}, τ_{j} ∈ C, and γ(x) := x_{2} . Then x_{1} and x_{2} are, up to multiplication with invertible series, the only irreducible semi-invariants of f in C[[x_{1}, x_{2}]]. If not all the σ_{j} are equal to zero then there exists no first integral that is algebraic over C((x_{1}, x_{2})). Up to multiplication by constants, the only integrating factor that is algebraic over C((x_{1}, x_{2})) is

(x_{1}x_{2}^{1+l}ρ(x))^{−1} = (x_{1}x_{2}γ(x)^{l}ρ(x))^{−1},

with l the smallest index such that σ_{l} ≠ 0, and ρ is defined similarly to (b).

The remaining case, with α_{2}/α_{1} rational and positive, is just the case of a dicritical nondegenerate stationary point, which is characterized by the existence of infinitely many irreducible semi-invariants; see [14], loc. cit., for details.

In [14], applications of (2.1) to polynomial vector fields were presented. Here we discuss an application to degenerate stationary points of an analytic differential equation ẋ = f(x). Degeneracy then means that B = Df(0) is nilpotent. The general properties of degenerate stationary points are not as well understood as those of nondegenerate ones. A relatively recent result by Camacho and Sad [5] shows that there is always an analytic semi-invariant. Little seems to be known about integrating factors.

A familiar strategy to deal with degenerate stationary points is to use blow-ups (σ -processes), and thus obtain stationary points of simpler type. One method in the real setting is to use polar coordinates (see Guckenheimer and Holmes [6], Ch. 7) to blow up a stationary point into an invariant circle that contains less degenerate stationary points. An other approach employs rational functions, and in essence yields a blow-up of a stationary point to a projective line; see Arnold [2], Ch. 1, §2, and also Anosov et al. [1], Ch. 5, §1, for details. According to theorems by Bendixson and Seidenberg (see [1], loc. cit.), every stationary point can be transformed to a collection of nondegenerate stationary points by a finite series of blow-ups. We will use the second version in an algebraic incarnation, which amounts to employing what Bruno [4] calls “power transformations”. In the following, a number of properties of such blow-ups, most of them well-known, will be listed, and ad-hoc proofs are provided for the sake of completeness. (There is no intention to state the results in full generality.)

Lemma2.2. Define Φ : C^{2} → C^{2} by

Φ(x) = ( \begin{matrix}x_{1} \\ x_{1}x_{2} \end{matrix} ) = ( \begin{matrix}x_{1} \\ x_{1}(x_{2} − a) + x_{1}a \end{matrix} ) ,

with a ∈ C arbitrary.

(a) For any a the map Φ induces a morphism from C[[x_{1}, x_{2}]] to C[[x_{1}, x_{2} −a]] via ψ ↦ ψ ∘ Φ, and convergent series are mapped to convergent series.

(b) For any a there is a unique analytic vector field f^{∗} near (0, a)^{t} such that the identity DΦ(x)f^{∗}(x) = f(Φ(x)) holds; thus Φ maps parameterized solutions of ẋ = f^{∗}(x) to parameterized solutions of ẋ = f(x). There is a unique integer s ≥ 0 such that f^{∗} = x_{1}^{s} · f̃, and x_{1} does not divide both entries of f̃.

(c) Let ψ be a formal or analytic semi-invariant of f, and let q be a nonnegative integer and ψ̃ a (formal resp. convergent) series such that (at the point (0, a)^{t}) one has ψ ∘ Φ = x_{1}^{q} · ψ̃, with x_{1} and ψ̃ relatively prime. Then ψ̃ is invertible or a semi-invariant of f̃.

(d) Let ρ be an irreducible local analytic semi-invariant of f̃ at (0, a)^{t}, with x_{1} and ρ relatively prime, and assume that locally the zero set Z of ρ is the image of some non-constant analytic map μ : U → C^{2} , with U ⊆ C open and nonempty. Then there is a semi-invariant σ of f such that Φ(Z) is contained in the zero set of σ. Proof. (a) For ψ = ∑ c_{ik}x_{1}^{i}x_{2}^{k}, the definitions and the binomial formula yield

ψ ∘ Φ = ∑_{i,k} c_{ik} ∑_{l=0}^{k} ( \begin{matrix}k \\ l \end{matrix} ) a^{l}x_{1}^{i+k}(x_{2} − a)^{l},

and clearly this series is summable, i.e., each monomial x_{1}^{m}(x_{2} − a)^{n} occurs only finitely many times with nonzero coefficient.

(b) Since f(0) = 0, each entry of f ∘ Φ is divided by x_{1} , whence the factor x_{1}^{−1} in the product with DΦ(x)^{−1} cancels. The second assertion is obvious.

(c) By a general argument, ψ^{∗} := ψ ∘Φ is a semi-invariant of f^{∗}. Since products and (non-invertible) divisors of semi-invariants are again semi-invariants, and x_{1} is a semi-invariant of f^{∗}, ψ̃ is also a semi-invariant of f^{∗}. Thus x_{1}^{s}Lf̃(ψ) = ν · ψ for some analytic ν, and x_{1}^{s} must divide ν. But this implies that ψ̃ is a semiinvariant for f̃.

(d) The image Φ(Z) = Φ(μ(U)) is contained in some one-dimensional local analytic set, which coincides with the local zero set of some analytic σ; cf. Ruiz [11]. Since Z is locally invariant for f̃ and for f^{∗}, Φ(Z) is locally invariant for ẋ = f(x), whence σ is a semi-invariant of f.

Remark2.3. (a) Obviously one gets analogous results for Θ(x) = \binom{x_{1}x_{2}}{x_{2}}, and for compositions of Φ and Θ.

(b) In the situation of (2.2)(c), ψ̃ can only be a semi-invariant (i.e., noninvertible) if (0, a)^{t} is a stationary point of f̃. If a nondegenerate stationary point of f̃ admits a linearization of a type as enumerated in (2.1), then the hypothesis of (2.2)(d) is satisfied, due to smoothness of the zero set of the semiinvariant and the implicit function theorem.

Let us now return to ẋ = f(x), with Df(0) = B. If B ≠ 0 then we may assume that B = ({0 1 \atop 0 0}), and this case will be discussed next. We will use a suitable normalization, and for our purposes the Arnold-Bogdanov normal form seems the most convenient. The first statement of the following lemma is wellknown (see, for instance, Guckenheimer and Holmes [6], 7.3), while the second results from an elementary computation.

Lemma2.4. Let B = ({0 1 \atop 0 0}), and f(x) = B(x) + ∑_{j≥2} f^{(j)}(x), with f^{(j)} homogeneous of degree j, a formal power series vector field. Then there exists an invertible formal transformation into Arnold-Bogdanov normal form

f^{∗}(x) = B(x) + ∑_{j≥2} ( \begin{matrix}0 \\ α_{j}x_{1}^{j−1}x_{2} + β_{j}x_{1}^{j} \end{matrix} ) .

If f is analytic then for any integer k ≥ 2 there is an analytic transformation into a vector field which is in normal form up to degree k. For

f(x) = B(x) + ( \begin{matrix}ρ_{1}x_{1}^{2} + ρ_{2}x_{1}x_{2} + ρ_{3}x_{2}^{2} \\ σ_{1}x_{1}^{2} + σ_{2}x_{1}x_{2} + σ_{3}x_{2}^{2} \end{matrix} ) + t. h. o.,

(with “t.h.o.” standing for “terms of higher order”), one has

f^{∗}(x) = B(x) + ( \begin{matrix}0 \\ σ_{1}x_{1}^{2} + (σ_{2} + 2ρ_{1})x_{1}x_{2} \end{matrix} ) + t. h. o.

Theorem2.5. The analytic equation

ẋ = f(x) := ( \begin{matrix}x_{2} \\ 0 \end{matrix} ) + ∑_{j=2}^{k} ( \begin{matrix}0 \\ α_{j}x_{1}^{j−1}x_{2} + β_{j}x_{1}^{j} \end{matrix} ) + t. h. o.,

with α_{2} ≠ 0 and β_{2} ≠ 0, and k ≥ 4, admits a unique irreducible analytic semi-invariant ϕ(x) = x_{2}^{2} + t. h. o. ∈ C[[x_{1}, x_{2}]], up to multiplication by an invertible series. Furthermore, if there is an integrating factor that is algebraic over C((x_{1}, x_{2})) then it is equal to ϕ^{−7/6} exp(μ), for some series μ which is unique up to an additive constant.

Proof. (i) To show the assertion about semi-invariants, we use a blow-up. We adopt the procedure in Bruno [4], Ch. II, 3.5. The map

Ψ(x) = ( \begin{matrix}x_{1}^{2}x_{2} \\ x_{1}^{3}x_{2}^{2} \end{matrix} )

is solution-preserving from ẋ = x_{1}f̃(x), with

f̃(x) = ( \begin{matrix}−β_{2}x_{1} + 2x_{1}x_{2} − α_{2}x_{1}^{2}x_{2} + … \\ 2β_{2}x_{2} − 3x_{2}^{2} + 2α_{2}x_{1}x_{2}^{2} + … \end{matrix} ) ,

to ẋ = f(x). (In the notation of (2.2) and (2.3) one has Ψ = Φ ∘ Θ ∘ Φ, hence the results above are applicable.) The inverse image of {0} under Ψ is the union Y of the coordinate axes. Therefore we have induced maps Ψ^{∗} = Ψ_{(a_{1}, 0)}^{∗} : C[[x_{1}, x_{2}]] → C[[x_{1} − a_{1}, x_{2}]], σ ↦ σ ∘ Ψ, and Ψ^{∗} = Ψ_{(0, a_{2})}^{∗} : C[[x_{1}, x_{2}]] → C[[x_{1}, x_{2} − a_{2}]], σ ↦ σ ∘ Ψ, for all a_{1}, a_{2} ∈ C. Also note that x_{1} and x_{2} are semi-invariants of f̃, or invertible, near any point of Y .

Let ψ be a semi-invariant of f. Then ψ ∘ Ψ is a semi-invariant of f̃ or invertible at any point of Y , and the same holds for the series ψ̃ obtained from ψ ∘ Ψ by dividing out maximal powers of x_{1} and x_{2} . Given a point (a_{1}, a_{2})^{t} on Y (so a_{1} = 0 or a_{2} = 0) such that the corresponding ψ̃ is not invertible, this point must be stationary for f̃. On Y there are the two stationary points (0, 0) and (0, \frac{2β_{2}}{3} ). (The higher order terms have no influence on this.) From Df̃(0, 0) = diag(−β_{2}, 2β_{2}) and (2.1) follows that x_{1} and x_{2} are the only irreducible semi-invariants for f̃ at (0, 0). Their vanishing sets are contained in Y , and therefore ρ̃ (taken at (0, 0)) is invertible for any semi-invariant ρ of f. The eigenvalues of Df̃(0, \frac{2β_{2}}{3} ) are \frac{β_{2}}{3} and −2β_{2} ; hence there are exactly two irreducible semi-invariants, x_{1} being one of them. The other one is easily computed as

ϕ̃:= 8α_{2}β_{2}x_{1} − 21(x_{2} − \frac{2β_{2}}{3} ) + t. h. o.

Thus, by (2.2)(c) there is only one irreducible (formal and analytic) semiinvariant ϕ for ẋ = f(x) at 0. To get its initial terms explicitly, one may proceed as in Bruno [4] , loc. cit., and use (2.2)(d). Computing the eigenvector of Df̃(0, 2β_{2}/3) yields a local analytic parameterization

v(s) = ( \begin{matrix}21s + … \\ 2β_{2}/3 + 8α_{2}β_{2}s + … \end{matrix} )

of the zero set of ϕ̃, and thus

Ψ(v(s)) = ( \begin{matrix}2β_{2}/3 · 21^{2}s^{2} + … \\ (2β_{2}/3)^{2} · 21^{3}s^{3} … \end{matrix} )

is a local analytic parameterization of the zero set of ϕ. This leads to

ϕ(x) = x_{2}^{2} − \frac{2}{3}β_{2}x_{1}^{3} + ….

(ii) Consider the equation L_{f}(ϕ) = d · div(f) · ϕ, for some rational d. With div(f) = α_{2}x_{1} + …, and ϕ = x_{2}^{2} + ϕ_{3} + … one obtains by comparing terms of degree 3 that

x_{2} \frac{∂ϕ_{3}}{∂x_{1}} + (α_{2}x_{1}x_{2} + β_{2}x_{1}^{2}) · 2x_{2} = d · α_{2}x_{1}x_{2}^{2},

which yields

ϕ_{3} = −\frac{2}{3}β_{2}x_{1}^{3} + (\frac{d}{2} − 1) α_{2}x_{1}^{2}x_{2} + ρ_{3}x_{2}^{3}.

Comparing homogeneous terms of degree 4 yields an equation x_{2} \frac{∂ϕ_{4}}{∂x_{1}} = …, and the condition that the coefficient of x_{1}^{4} must vanish in the remaining terms turns out to force (1 − \frac{d}{2}) α_{2}β_{2} = \frac{2}{3}dα_{2}β_{2} , hence d = \frac{6}{7} . (iii) The equation admits no (nonconstant) first integral in C((x_{1}, x_{2})). Such a first integral σ would be a product of powers of irreducible semi-invariants and an invertible series. Since ϕ is the only irreducible semi-invariant, this would imply that σ = ϕ^{k} exp(ν). One may assume that k = 1 (take some power of σ, which is also a first integral, and note that σ(0) = 0 w.l.o.g.). But now the result of (ii), with d = 0, yields a contradiction. As a consequence, there is no first integral which is algebraic over C((x_{1}, x_{2})), since the coefficients of its minimum polynomial would themselves be first integrals or constant.

From (ii) it also follows that L_{f}(ϕ exp(μ))^{1/d} = div(f)(ϕ exp(μ))^{1/d} forces d = \frac{6}{7} , and the integrating factor is unique according to the previous paragraph.

Corollary2.6. (a) If the vector field

f(x) = B(x) + ( \begin{matrix}ρ_{1}x_{1}^{2} + ρ_{2}x_{1}x_{2} + ρ_{3}x_{2}^{2} \\ σ_{1}x_{1}^{2} + σ_{2}x_{1}x_{2} + σ_{3}x_{2}^{2} \end{matrix} ) + t. h. o.,

with σ_{1} ≠ 0, and 2ρ_{1} + σ_{2} ≠ 0, admits an integrating factor that is algebraic over C((x_{1}, x_{2})) then it is unique and of the form (x_{2}^{2} + t. h. o.)^{−7/6} .

(b) In the setting of (2.5) there exists an integrating factor only if a certain polynomial in the α_{i} and β_{i}, with i ≤ 3, vanishes. Thus, in general there will be no algebraic integrating factor.

Proof. (Sketch.) Part (a) is immediate from (2.4) and (2.5), while part (b) follows from a straightforward investigation of the degree 5 and degree 6 terms of the recursion in part (ii) of the proof of (2.5).

Remark2.7. As (2.6) shows, computation of a normal form with respect to a nilpotent linear part will not generally produce a vector field that is integrable in an elementary manner. Note that this fact does not depend on the specific choice of a normal form. This stands in marked contrast to the case of two-dimensional normal forms with respect to a semisimple linear part. However, there do exist vector fields which admit an integrating factor as in (2.6). For instance, the equation

ẋ = ( \begin{matrix}x_{2} \\ 0 \end{matrix} ) + ( \begin{matrix}2x_{1}^{2} \\ \frac{3}{2}x_{1}^{2} + 7x_{1}x_{2} \end{matrix} )

(which was constructed using [13], Prop. (3.4), with B = diag(2, 3)), admits the integrating factor (x_{1}^{3} − x_{2}^{2})^{−7/6} . It is worth noting that the irreducible semiinvariant of an equation as in (2.6), if it exists, is formally equivalent to x_{2}^{2} +τx_{1}^{3} , for some τ ≠ 0. (This follows from the expression for ϕ_{3} in the proof of (2.4), and a Newton polygon argument; see, for instance, Bruno [4], Ch. I.) Therefore [13] essentially allows the construction of all the equations in (2.6) which admit an algebraic integrating factor.

Let us next discuss degenerate stationary points with vanishing linearization. We do not attempt to investigate these completely, but we will present some results showing that generically there will be no integrating factor which is algebraic over C((x_{1}, x_{2})). Thus, let ẋ = f(x) be given, with f ≠ 0, f(0) = 0, and Df(0) = 0. Then there is an integer r ≥ 2 such that f = ∑_{k≥r} f^{(k)} , with homogeneous polynomials f^{(k)} , and f^{(r)} ≠ 0. Let us collect a few well-known facts about homogeneous systems. (One source of complete proofs is [15].)

Lemma2.8. For a given homogeneous polynomial vector field f^{(r)} , define ψ(x) := det(f^{(r)}(x), x). Then ψ ≠ 0 unless there is a polynomial μ such that f(x) = μ(x) · x for all x.

If ψ ≠ 0 then L_{f^{(r)}}(ψ) = div(f^{(r)})ψ, hence ψ^{−1} is an integrating factor. Moreover, ψ = ϕ_{1} ⋯ ϕ_{r+1} is a product of linear forms, and there are homogeneous polynomials λ_{i}, of degree r − 1, such that L_{f^{(r)}}(ϕ_{i}) = λ_{i}ϕ_{i}, for 1 ≤ i ≤ r + 1.

The vector field f^{(r)} admits a (homogeneous) rational first integral if and only if the λ_{i} are linearly dependent over the field of rational numbers.

Proof. (Sketch.) It follows from [13], Prop. 1.1, that ψ^{−1} is an integrating factor. Since the only irreducible complex homogeneous polynomials in two variables are those of degree 1, one gets a prime factorization with linear forms. Since ψ is semi-invariant, so are all the ϕ_{i}.

As to the last assertion, note that if there is a rational first integral then there is a homogeneous rational first integral, which must be of the form ϕ_{1}^{k_{1}} ⋯ ϕ_{r+1}^{k_{r+1}} . (This follows from simple computations; see [15].) The observation

L_{f^{(r)}}(ϕ_{1}^{k_{1}} ⋯ ϕ_{r+1}^{k_{r+1}}) = (k_{1}λ_{1} + … + k_{r+1}λ_{r+1})(ϕ_{1}^{k_{1}} ⋯ ϕ_{r+1}^{k_{r+1}})

then finishes the proof.

For the detailed investigation of the stationary point a blow-up is once again useful.

Lemma2.9. Let the analytic differential equation ẋ = f(x) be given, with f = ∑_{k≥r} f^{(k)} , and

f^{(r)} = ( \begin{matrix}∑_{i=0}^{r} ρ_{i}x_{1}^{r−i}x_{2}^{i} \\ ∑_{i=0}^{r} σ_{i}x_{1}^{r−i}x_{2}^{i} \end{matrix} ) .

Assume that ρ_{r} ≠ 0 (whence ψ(x) := det(x, f^{(r)}(x)) is not identically zero), and that ψ has pairwise relatively prime linear factors ϕ_{1}, …, ϕ_{r+1} .

Then Φ(x) = ({x_{1} \atop x_{1}x_{2}} ) is solution-preserving from an analytic equation ẋ = x_{1}^{r−1}f̃(x) to ẋ = f(x), with

f̃(x) = ( \begin{matrix}ρ_{0}x_{1} + ρ_{1}x_{1}x_{2} + x_{1}^{2}(…) \\ ν(x_{2}) + x_{1}(…) \end{matrix} ) ,

and ν(x_{2}) = σ_{0} + (σ_{1} − ρ_{0})x_{2} + … + (σ_{r} − ρ_{r−1})x_{2}^{r} − ρ_{r}x_{2}^{r+1} .

The vector field f̃ has exactly r + 1 stationary points on the invariant line x_{1} = 0, and these are nondegenerate. If none of these is dicritical then f has precisely r + 1 irreducible analytic (or formal) semi-invariants ψ_{i} = ϕ_{i} + t. h. o. in 0.

Proof. This follows immediately from (2.2) and (2.3), since by (2.1) every stationary point on x_{1} = 0 yields precisely one nontrivial semi-invariant.

Theorem2.10. Let the notation and hypotheses be as in (2.9). Assume that none of the stationary points of f̃ on the line x_{1} = 0 is dicritical, and that f^{(r)} has no nonconstant rational first integral.

(a) If ẋ = f(x) admits an integrating factor that is algebraic over C((x_{1}, x_{2})) then it is necessarily of the form (ϕ_{1} ⋯ ϕ_{r+1} + t. h. o.)^{−1} .

(b) Given f^{(r)} , the vector field f^{(r+1)} can be chosen in such a way that ẋ = f(x) admits no algebraic integrating factor.

Proof. (a) Assume that f admits the integrating factor

(ψ_{1}^{d_{1}} ⋯ ψ_{r+1}^{d_{r+1}})^{−1} exp(μ).

Then a simple computation shows that f^{(r)} admits the integrating factor

(ϕ_{1}^{d_{1}} ⋯ ϕ_{r+1}^{d_{r+1}})^{−1},

and (2.7) shows that d_{1} = … = d_{r+1} = 1.

(b) Let γ^{−1} be an integrating factor for f. Then γ = ∑_{i≥r+1} γ_{i}, with γ_{i} a homogeneous polynomial of degree i, and γ_{r+1} ≠ 0 is a product of pairwise different primes, according to part (a). The integrating factor condition yields

(∗)  L_{f^{(r)}}(γ_{r+2}) − div(f^{(r)})γ_{r+2} = −L_{f^{(r+1)}}(γ_{r+1}) + div(f^{(r+1)})γ_{r+1}.

Denote by S_{j} the vector space of homogeneous polynomials of degree j (which has dimension j + 1), and by P_{j} the space of homogeneous vector fields with entries of degree j.

The image of the linear map Δ : S_{r+2} → S_{2r+1} , μ ↦ L_{f^{(r)}}(μ) − div(f^{(r)})μ, has dimension at most equal to r + 3 < 2r + 2, and therefore Δ is not onto.

On the other hand, the linear map Γ : P_{r+1} → S_{2r+1} , p ↦ L_{p}(γ_{r+1}) − div(p)γ_{r+1} , is onto: Note that p ∈ ker(Γ) if and only if γ_{r+1}^{−1} is an integrating factor for p. Since the entries of γ_{r+1}^{−1}p are homogeneous of degree 0, it follows from the proof of [13], Prop. 3.4, that this is the case if and only if

p = ρ ( \begin{matrix}−∂γ_{r+1}/∂x_{2} \\ ∂γ_{r+1}/∂x_{1} \end{matrix} ) + λx,

with a linear form ρ, and λ (and therefore p) uniquely determined by ρ. This shows that dim(ker(Γ)) = dim(S_{1}) = 2, and comparing dimensions proves the surjectivity of Γ.

Therefore, the equation Δ(γ_{r+2}) = −Γ(f^{(r+1)}) (which is just (∗) rewritten) has a solution only if f^{(r+1)} lies in some fixed proper subspace of P_{r+1} .

Note that the question of existence of rational first integrals of f^{(r)} can be decided by (2.8). One also knows that there can be no rational first integral of f^{(r)} if one of the stationary points of f̃ on the line x_{1} = 0 has a non-rational eigenvalue ratio; see (2.1).

If f^{(r)} is given then an elementary computation yields the conditions on the coefficients of f^{(r+1)} . Moreover, there will be additional conditions from higher degree terms. For example, the equation

ẋ = ( \begin{matrix}−2x_{1}^{2} + x_{1}x_{2} + 2x_{2}^{2} \\ −x_{1}x_{2} + x_{2}^{2} \end{matrix} ) + ( \begin{matrix}x_{1}x_{2}^{2} \\ 0 \end{matrix} )

admits no integrating factor that is algebraic over C((x_{1}, x_{2})).

In view of the theorems of Bendixson and Seidenberg, the strategy of blowing up and reducing to the nondegenerate case is clearly applicable to all degenerate stationary points in the plane. But with the currently available methods one seems to be restricted to a case-by-case analysis of various special classes.

References

[1] Anosov, D. V. et al., “ Ordinary differential equations and smooth dynamical systems,” Springer-Verlag, Berlin etc., 1997.

[2] Arnold, V. I., “ Geometrical methods in the theory of ordinary differential equations,” Springer-Verlag, Berlin etc., 1988.

[3] Bibikov, Yu. N., “Local theory of analytic ordinary differential equations,” Lecture Notes in Mathematics 702. Springer-Verlag, Berlin etc., 1979.

[4] Bruno, A. D., “ Local methods in nonlinear differential equations,” Springer-Verlag, Berlin etc., 1989.

[5] Camacho, C., and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. Math. 115 (1982), 579–595.

[6] Guckenheimer, J., and P. Holmes, “Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,” Springer-Verlag, Berlin etc., 1983.

[7] Lie, S., ,,Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen,“ Teubner-Verlag, Leipzig 1891; Reprint Chelsea Publ., 1967.

[8] Olver, P. J., “Applications of Lie groups to differential equations,” Springer-Verlag, New York etc., 1986.

[9] —, “Equivalence, invariants and symmetry,” Cambridge University Press, 1995.

[10] Prelle, M. J., and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1982), 215–229.

[11] Ruiz, J. M., “The basic theory of power series,” Vieweg-Verlag, Braunschweig, 1993.

[12] Walcher, S., On differential equations in normal form, Math. Annalen 291 (1991), 293–314.

[13] —, Plane polynomial vector fields with prescribed invariant curves, Proc. Royal Soc. Edinburgh 130A (2000), 633–649.

[14]  —, On the Poincaré problem, J. Diff. Eqs. 166 (2000), 51–78.
[15]  —, “Algebras and differential equations,” Hadronic Press, 1991.

Sebastian Walcher Lehrstuhl A für Mathematik RWTH Aachen 52056 Aachen Germany walcher@mathA.rwth-aachen.de

Received April 23, 2002 and in final form September 10, 2002