RESEARCH ARTICLE Newton Vector Fields on the ...

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Vol. 00, No. 00, July 2008, 1–13

RESEARCH ARTICLE Newton Vector Fields on the Plane and on the Torus Alvaro Alvarez-Parrillaa∗ , Adrian G´ omez-Arcigaa, and Alberto Riesgo-Tiradoa a Department of Mathematics, Facultad de Ciencias, Universidad Aut´ onoma de Baja California, B.C., M´exico (v1.0 released soon in 2008) In this paper we show that under certain conditions, complex vector ﬁelds on the punctured Riemann sphere are Newton vector ﬁelds and as a consequence all meromorphic vector ﬁelds and all elliptic vector ﬁelds are Newton vector ﬁelds. Moreover for a large class of vector ﬁelds, which includes all elliptic vector ﬁelds, the proof is constructive in the sense that one can explicitly construct the function characterising the Newton vector ﬁeld.

Keywords: Meromorphic and elliptic Newton ﬂows; pullback vector ﬁelds; branched covering maps; Mittag-Leﬄer expansion; Weirstrass ℘-function. AMS Subject Classification: 37F75; 32M25; 37F10; 33E05; 30D30; 33E20

1.

Introduction

Vector ﬁelds have been extensively studied in relation to many distinct problems and from several points of view [See 1–15, and references therein]. We could mention that in the case of analytic vector ﬁelds of one complex variable, there is a correspondence between analytic vector ﬁelds, certain autonomous diﬀerential equations, meromorphic diﬀerential forms, and Riemannian metrics arising from these diﬀerential forms [2, 10, 11]. Of particular interest are the so called Newtonian (or Newton) vector ﬁelds which are vector ﬁelds which can be expressed as X(z) = −

Ψ(z) ∂ , Ψ (z) ∂z

for some function Ψ(z). There has been notable interest in Newtonian vector ﬁelds, since the 80’s when Hirsch, Smale and others [see 9, 14, 15, and references therein], studied the so called Newton ﬂow of a polynomial in relation to complexity issues associated to Newton’s method of root ﬁnding. Since then much has been done regarding Newton ﬂows and Newton vector ﬁelds, but we wish to call attention to the work of Jongen et al. [6, 7], Helminck et al. [8], and Benzinger [3, 4]: In [6, 7] a complete characterisation/classiﬁcation (up to conjugacy) of rational Newton ﬂows is undertaken under mild generic conditions. In [8] they show that the Newton ﬂow associated to Weirstrass’ ℘-function can ∗ Corresponding

author. Email: [email protected]

ISSN: print/ISSN online c 2008 DOI: http://alvaro.uabc.googlepages.com

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be characterised/classiﬁed (again up to conjugacy) into three types of behaviour according to whether the fundamental parallelogram associated to ℘ is square, rectangular but not square, and non-rectangular. In [3] Benzinger shows that all rational vector ﬁelds are Newton vector ﬁelds 1 and with this information proceeds to show that the Riemann sphere is decomposed into a ﬁnite collection of open sets where the ﬂow is analytic and where there is common long time behaviour, and in [4] he shows that the Julia set of z + hR(z), for small h, is related to the phase plane portrait of the continuous dynamical system z˙ = R(z), where R(z) is an arbitrary rational function (and hence to the Newton ﬂow associated to R(z)). As can be seen in the works mentioned above [3, 4, 6–8], the fact that the associated vector ﬁelds are Newton vector ﬁelds provides the necessary framework and tools to prove their results. Hence knowing that a family of vector ﬁelds are Newton vector ﬁelds is a big step in understanding the behaviour of said family of vector ﬁelds. In line with this last remark, the ﬁrst author of the present work has shown that a method, ﬁrst presented in [3] for visualising rational vector ﬁelds, can be extended to all Newton vector ﬁelds once the function Ψ(z) is explicitly known. It should be noted that this new visualisation method has clear advantages over the traditional ones, mainly that there is no need for an iteration procedure (hence there is no error propagation) and that information about the parametrization of the solutions is easily accessible [1]. In this work we show that, under certain conditions, complex vector ﬁelds on the punctured Riemann sphere are Newton vector ﬁelds (Theorem 3.1) and as a consequence all meromorphic vector ﬁelds and all elliptic vector ﬁelds are Newtonian, moreover we show that in some large families of said vector ﬁelds (all elliptic vector ﬁelds and a large family of meromorphic vector ﬁelds) the function Ψ(z) can be explicitly calculated. In order to do this we present a geometrical framework where ∂ it is shown that Newton vector ﬁelds, X(z) = − ΨΨ(z) (z) ∂z , are in fact pullbacks of ∂ via precisely the function Ψ(z), which turns out to the vector ﬁeld Y (w) = −w ∂w be a branched cover of the sphere onto itself. The layout of the paper is as follows: In section §2 we provide the necessary background, namely deﬁne the concepts of complex analytic vector ﬁelds, branched coverings of the sphere, pullbacks of complex analytic vector ﬁelds via branched coverings, and Newton vector ﬁelds. In section §3 we proceed to show that complex analytic vector ﬁelds on (meromorphic on ) are Newtonian, then in §3.1 we show that for some families of said vector ﬁelds we can explicitly obtain the covering map Ψ, and in §3.2 we also show that doubly periodic (elliptic) vector ﬁelds are Newtonian vector ﬁelds, again by explicitly constructing the covering map. Finally in §4 we indicate some lines for future research.

2.

Background

In this section we deﬁne the concepts needed and recall some other concepts from the literature. Namely we deﬁne complex analytic vector ﬁelds, branched coverings of the sphere, pullback vector ﬁelds, and Newtonian vector ﬁelds.

1 In fact Benzinger studies the solutions to the diﬀerential equation z˙ = R(z), where R is any rational ∂ . function, but this is equivalent to studying the rational vector ﬁelds R(z) ∂z

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Complex Analytic Vector Fields

In this section we deﬁne the main objects of study, namely complex analytic vector ﬁelds. Definition 2.1: Suppose f (z) = u(z)+iv(z) is a meromorphic function on U ⊂ , then X(z) = f (z)

∂ ∂ = u(z) + iv(z) , ∂z ∂z

is called a complex analytic vector ﬁeld on U ⊂ . Given a complex analytic vector ﬁeld one has two real valued vector ﬁelds given by 2Re(X)(x, y) = F (x, y) = u(x, y)

∂ ∂ + v(x, y) , ∂x ∂y

and −2Im(X)(x, y) = JF (x, y) = −v(x, y)

∂ ∂ + u(x, y) . ∂x ∂y

It is convenient to note that these real vector ﬁelds have the characteristic of being perpendicular everywhere, so it is suﬃcient to know only one of them in order to completely specify the complex analytic vector ﬁeld. Moreover, each of the real vector ﬁelds has a real ﬂow ψ associated to it, and in the case of the vector ﬁeld F it is deﬁned by

ψz 0 (t) = F (ψz0 (t)) ψz0 (0) = z0 .

(1)

By convention when we speak of the trajectories or streamlines of the complex vector ﬁeld X, we will be referring to the the ﬂows (streamlines) of the associated real vector ﬁelds, in particular unless otherwise noted we will be referring to the ﬂows of F .

2.2.

Branched Coverings of the Sphere

Definition 2.2: A covering Ψ : V → W is a continuous surjective mapping such that ∀w ∈ W, ∃ an open set U w in W with the characteristic that Ψ−1 (U ) is a disjoint union of open sets O ⊂ V each of which satisﬁes that Ψ : O → U is a homeomorphism. A branched or ramiﬁed covering Ψ : V → W is a covering except at a ﬁnite number of points of W . The exceptional points are known as branch points or ramiﬁcation points. Note that if one has a complex analytic vector ﬁeld X(z) on then it is natural to think of X(z) as a vector ﬁeld on the Riemann sphere = ∪ {∞}. Note that ∞ might not necessarily be an isolated singularity. In any case we will consider X(z) as a vector ﬁeld on or on as we see ﬁt.

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2.3.

Pullback of Complex Analytic Vector Fields

Usually when studying complex vector ﬁelds and their associated real ﬂows, one tries to solve the system of diﬀerential equations given by φz0 (t) = f (φz0 (t)) (2) φz0 (0) = z0 , However, recently the ﬁrst author and Muci˜ no-Raymundo [2] have studied complex vector ﬁelds in a neighbourhood of their zeros and singularities with the use of the concept of pullback of a vector ﬁeld: that is given a complex vector ﬁeld Y (w) deﬁned on w and a map Ψ : z → w analytic at z such that w = Ψ(z) with Ψ (z) = 0, then tangent vectors at z are taken to tangent vectors at w via the derivative of Ψ. In particular if Ψ is an analytic branched covering of the sphere then one can consider the following. Definition 2.3: The pullback of Y via Ψ is the complex vector ﬁeld X(z) on z whose image by Ψ is Y (w). The relation between a complex analytic vector ﬁeld and its pullback is given by the following. ∂ is a complex vector ﬁeld on w the analytic Lemma 2.4: If Y (w) = g(w) ∂w ∂ , called the pullbranched covering Ψ deﬁnes a complex vector ﬁeld X(z) = f (z) ∂z ∗ back of Y via Ψ, denoted by X = Ψ (Y ) and the relationship between X and Y is given by

f (z) =

g(Ψ(z)) . Ψ (z)

Proof : The proof is an immediate consequence of the chain rule.

As an important example, consider pullbacks by Fractional Linear Transformations (bi-holomorphisms of the sphere): ∂ be a Example 2.5 Fractional Linear Transformations. Let Y (t) = g(t) ∂t complex vector ﬁeld on t and consider the pullback via a bi-holomorphism of

T : z → t z →

az + b , cz + d

with ad − bc = 0, then ∂ (cz + d)2 g(T (z)) . T ∗ Y (t) (z) = (ad − bc) ∂z 2.4.

Newtonian Vector Fields

In the 80’s Smale [9, 14, 15] introduced the concept of Newtonian ﬁeld in the deﬁnition of Newtonian graphs that arise from studying Newton’s method of root ﬁnding for a complex polynomial. Taking his deﬁnition as a guide we have:

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∂ Definition 2.6: A complex analytic vector ﬁeld X(z) = f (z) ∂z is said to be a Newtonian vector ﬁeld (or Newton vector ﬁeld) on U ⊂ if it can be represented as

X(z) = −

Ψ(z) ∂ , Ψ (z) ∂z

for some analytic (possibly multivalued) function Ψ on U . Remark 1 : Note that if Ψ (z0 ) = 0 then X(z) either has a pole or is regular at z0 , so X(z) is still a complex analytic vector ﬁeld on U , for simplicity we still say that X(z) is Newtonian at z0 . An immediate consequence are the following equivalencies. Lemma 2.7:

The following are equivalent:

∂ is a Newtonian ﬁeld (1) A vector ﬁeld X(z) = f (z) ∂z on U . z 1 Ψ(z) ∂ (2) X(z) = − Ψ (z) ∂z with Ψ(z) = exp − f (ζ) dζ analytic (possibly mul-

tivalued) on U , where the integral is over a C 1 path γ(t) contained in U such that γ(0) = z0 ﬁxed1 and γ(1) = z. ∂ via Ψ. (3) X(z) is the pullback of −w ∂w Proof : The proof of (1) ⇔ (3) is immediate from the deﬁnition of pullback vector ﬁeld and from Lemma 2.4. To obtain the equivalence (1) ⇔ (2), consider Ψ (z) ∂ 1 =− =− log Ψ(z) , f (z) Ψ(z) ∂z

(3)

integrate along a C 1 path γ in U for some z0 ∈ U chosen so that (3) is satisﬁed, and exponentiate. Remark 2 (Geometrical interpretation of the lemma): The geometrical interpretation of the previous Lemma can be understood in terms of the pullback as follows: The trajectory z(t) of a Newton vector ﬁeld X(z), is the ﬂow of the pullback, ∂ (whose trajectories are straight lines, parametrized by e−t , via Ψ, of the ﬁeld −w ∂w that start at ∞ and end at 0 in w ). Moreover Ψ (z(t)) = Ψ(z0 )e−t , ∀t ∈ . Refer to Fig. 1.

↓ Ψ

∂ Figure 1. The trajectories z(t) correspond to trajectories of −t ∂t under the covering map Ψ.

1 The

choice of z0 ∈ U is determined by the function f (z) in order to satisfy (3).

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3.

Newtonian Vector Fields on the punctured sphere

In this section we state and prove the main theorems, that is we show that certain families of vector ﬁelds on the punctured sphere are Newtonian. In particular, we show in §3.1 that all complex analytic vector ﬁelds on are Newtonian, and that for a large class of complex analytic vector ﬁelds on the plane one can explicitly ﬁnd an expression for the branched covering Ψ(z). In §3.2 we show that all doubly periodic meromorphic (elliptic) vector ﬁelds on the plane are Newtonian, again explicitly showing an expression for the covering map Ψ(z). ∂ be a complex vector ﬁeld on , and let S ⊂ Theorem 3.1 : Let X(z) = f (z) ∂z be acountable set of points having a ﬁnite number of accumulation points. Ψ(z) = z 1 exp − f is an analytic (possibly multivalued) function on \S if and only if X(z) = f (z) ∂ is a Newtonian vector field on \S. ∂z

Proof : Without loss of generality we may assume that one of the accumulation points of S is ∞: otherwise let s be an accumulation point of S, and consider the ∂

, where Ts (w) = sw−b is a pullback vector ﬁeld X(w) = Ts∗ (X(z)) (w) = f (w) ∂w w sw−b 2

Fractional Linear Transformation such that Ts (∞) = s, and f (w) = w f w .

where S = T −1 (S), and

Note then that X(w) is a complex vector ﬁeld on \S, s z dζ w dζ moreover f (ζ) = e . So in fact one may assume that one of the accumulation f (ζ) points of S is ∞, as stated. z 1 Now considering Ψ(z) = exp − f for z ∈ \S, the result follows by a direct application of Lemma 2.7. As an example of a vector ﬁeld for which the above theorem applies consider ∂ , then it is easily checked that Ψ(z) = exp (exp(−z)). It should X(z) = exp(z) ∂z be remarked that this example is interesting in that it is a vector ﬁeld with an essential singularity (at ∞ ∈ ). For more examples constructed in an ad-hoc manner consult §4.2. of [1]. Remark 1 : Note that the class of vector ﬁelds that are Newtonian includes those which have essential singularities.

3.1.

Meromorphic vector fields

As a ﬁrst application of Theorem 3.1, we show, using the classical Mittag-Leﬄer’s theorem on the characterisation of meromorphic functions on the plane, that all ∂ has meromorphic vector ﬁelds (i.e. those for which the vector ﬁeld X(z) = f (z) ∂z f (z) as a meromorphic function on ) are in fact Newtonian. We start by recalling [see Corollary after Theorem 10.10 on page 301 of 16, or any other classical treaty on complex variables for a proof]. Theorem 3.2 (Mittag-Leﬄer): Let g(z) be a meromorphic function on , and let P denote the set of poles of g, then g(z) = G(z) +

Gp (z) + Pp (z) ,

(4)

p∈P

where G(z) is an entire function, Gp (z) is the principal part of g at the pole p, and Pp (z) are polynomials such that P0 = 0 and that make the series converge uniformly on compact sets of \P.

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Mittag-Leﬄer’s theorem together with Theorem 3.1 provides us with: ∂ with f meromorphic on Corollary 3.3: Let X(z) = f (z) ∂z Ψ(z) ∂ Newtonian vector ﬁeld X(z) = − Ψ (z) ∂z , with

 Ψ(z) = exp −

 z  

G(ζ) +

p∈P

, then X(z) is a

  Gp (ζ) + Pp (ζ)  , 

where the integrand is the Mittag-Leﬄer expansion of g = poles.

1 f

and P is its set of

∂ with f (z) meromorphic Proof : Consider the complex vector ﬁeld X(z) = f (z) ∂z 1 on . Since g = f is also meromorphic on , then by using the Mittag-Leﬄer’s expansion of g we obtain (4), hence by uniform convergence of said series, the integral

z

g(ζ)dζ =

z

G(ζ)dζ +

z

Gp (ζ)dζ +

z

Pp (ζ)dζ ,

(5)

p∈P

is an analytic (possibly multivalued) function. Let us consider in slightly better detail the that arise: z diﬀerent integrals z G(ζ)dζ and Pp (ζ)dζ are also entire Since G(ζ) and Pp (ζ) are entire, then functions. The integrals of the principal parts

Gp (ζ) =

kp

p aj

j=1

(z − p)j

,

where p ∈ P is a pole of g and kp ≥ 1 denotes its order, give rise to terms of two diﬀerent forms, j = 1: in which case the (indeﬁnite) integral is log (z − p)Rp , where Rp = p a−1 is the residue of g at p, −p a−j j > 1: in which case the (indeﬁnite) integral of each term is (j−1)(z−p) j−1 . Finally, by Theorem 3.1 the (branched) covering Ψ is given by Ψ(z) = exp −

z

g(ζ)dζ ,

so in fact Ψ(z) = C exp −

z

G(ζ)dζ

 kp −Rp  (z − p) exp

p∈P

j=2

where C is an integration constant.

p a−j − (j − 1)(z − p)j−1

 z

Pp (ζ)dζ  ,

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Note that if X(z) has a zero at p, then the corresponding 1-form g(ζ)dζ has a pole at p, and if the residue at p, Rp is not an integer then Ψ(z) is a multivalued map. Moreover if the Mittag-Leﬄer expansion of f1 is known in terms of functions which have explicit antiderivatives, then the covering Ψ can be expressed in terms of well known functions. As a ﬁrst example of this situation we recall Cauchy’s Theorem on Partial Fractions [see Theorem 9.2.7, page 56 of 16, for a proof]. Theorem 3.4 Cauchy’s Theorem on Partial Fraction Expansions : Let g(z) be an analytic function whose only singular points in the ﬁnite plane are poles, and let {Ln } be a sequence of closed rectiﬁable Jordan curves with the following properties: (1) (2) (3) (4)

The origin lies inside each curve Ln (n = 1, 2, . . . ); None of the curves passes through poles of g(z); Each curve in the sequence is contained in the next, i.e., Ln ⊂ int(Ln+1 ); If rn is the distance from the origin to Ln , then lim rn = ∞.

n→∞

Let the poles of g(z) be ordered in such a way that Ln contains the ﬁrst mn + 1 poles b0 = 0, b1 , . . . , bmn , with principal parts1 G0 (z), G1 (z), . . . , Gmn (z), so that mn ≤ mn+1 (n = 1, 2, . . . ). Moreover, suppose that lim sup n→∞

Ln

|g(ζ)| ds = M < ∞ |ζ|p+1

for some integer p ≥ −1, where ds = |dζ|, and let z be an arbitrary regular point of g(z). Then, if p = −1, mn

Gk (z),

(6)

Gk (z) + Pk (z) ,

(7)

g(z) = lim

n→∞

k=0

while, if p > −1, g(z) = lim

n→∞

mn k=0

where the Pk (z) are polynomials of degree no higher than p, and the convergence in both (6) and (7) is uniform on every compact set containing no poles of g(z). Let F be the family of functions that satisfy the requirements of Cauchy’s Theorem on Partial Fractions. Denote by

Gk (z) =

Nk j=1

1 If

0 is not a pole then G0 (z) = 0.

ajk , (z − bk )j

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the principal part of g(z) at the pole bk of order Nk ; and let Pk (z) =

p

ajk z j ,

j=0

denote the corresponding polynomials (in case p > −1). Then we have: Theorem 3.5 : Let g(z) ∈ F then there is a meromorphic (possibly multivalued) function Ψ(z) on such that g(z) = −

Ψ (z) , Ψ(z)

and in fact Ψ(z) =

∞

“

(z − bk )−Ak e

qk

1 z−bk

”

eQk (z) ,

(8)

k=1

where qk (z) and Qk (z) are unique polynomials with qk (0) = 0 and Qk (0) = 0. Proof : By Cauchy’s Theorem on Partial Fractions we have that g(z) = lim

n→∞

mn

[Gk (z) + Pk (z)] ,

(9)

k=0

where in case p = −1 then Pk = 0. Hence

  p mn Nk Ψ (z) a jk  = − lim +

ajk z j  , log(Ψ) = n→∞ Ψ(z) (z − bk )j j=1

k=0

(10)

j=0

so that by integrating we obtain log Ψ(z) = − lim

n→∞

mn

 log(z − bk )a1k

 p Nk ajk

ajk j+1  1 z − + , j−1 j − 1 (z − bk ) j+1 j=2

k=0

j=0

(11) and by exponentiating and renaming Ak = a1k we have Ψ(z) = lim

n→∞

where qk (z) =

Nk j=2

ajk j−1 , j−1 z

mn

(z − bk )−Ak e

“

qk

1 z−bk

”

eQk (z) ,

(12)

k=1

and Qk (z) = −

p j=0

Corollary 3.6:

∂ Let X(z) = f (z) ∂z with

vector ﬁeld X(z) =

∂ − ΨΨ(z) (z) ∂z ,

e ajk j+1 . j+1 z

1 f

∈ F , then X(z) is a Newtonian

with Ψ(z) as in Theorem 3.5.

Remark 2 : It should be remarked that the family of functions F includes all the rational functions, and some transcendental functions that have ∞ ∈ as an accumulation point [for example tan(z) and cot(z) as can be seen in §10 of 16].

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3.2.

Newtonian Vector Fields on the Torus

In this section we show that all vector ﬁelds that are doubly periodic and meromorphic on the plane (elliptic) are Newtonian. Since elliptic functions can be identiﬁed with functions on the torus we have as a consequence that complex analytic vector ﬁelds on the torus (without essential singularities) are Newtonian. For an in depth review of the concepts associated to elliptic functions we refer the reader to [16] or any other classical treatment of the subject, we only state the main results we are to use. We start by recalling some well known results regarding elliptic functions. The ﬁrst one [see Theorem 25.5.12, page 181 of 16, for a proof], tells us that elliptic functions “behave like rational functions”: Theorem 3.7 : Given an elliptic function f (z) of order n, with fundamental periods 2w1 and 2w3 , let a1 , ..., an and b1 , ..., bn be the zeros and poles of f (z) in the fundamental period parallelogram, each counted a number of times equal to its order. Then f (z) = C

σ(z − a1 )σ(z − a2 ) · · · σ(z − an ) , σ(z − b1 )σ(z − b2 ) · · · σ(z − bn )

where σ(z) is Weirstrass’ sigma function, C is a constant and bn = (a1 + · · · + an ) − (b1 + · · · + bn−1 ). A direct consequence is that elliptic functions form a ﬁeld, the inverse of f (z) being 1 σ(z − b1 )σ(z − b2 ) · · · σ(z − bn ) 1 = . f (z) C σ(z − a1 )σ(z − a2 ) · · · σ(z − an ) 1 is also an elliptic function (in the proof of the above theorem this is Moreover f (z) shown). Next we recall that elliptic functions have a “partial fraction decomposition” in the same way that rational functions do [see Theorem 25.5.13 page 182 of 16, for a proof]:

Theorem 3.8 (Partial fraction decomposition for elliptic functions) : Given an elliptic function f(z) with fundamental periods 2w1 and 2w3 , let b1 , ..., br be the poles of f(z) in the fundamental period parallelogram. Suppose bk is of order βk , with principal part Gk =

Aβk k A1k + ··· + z − bk (z − bk )βk

(k = 1, ..., r).

Then βk r Ajk ζ (j−1) (z − bk ), (−1)j−1 f (z) = C + (j − 1)! k=1 j=1

where ζ(z) is Weirstrass’ zeta function and C is a constant. Remark 3 : Since ζ (z) = ℘(z), then one may replace ζ (j) by ℘(j−1) for j > 0, where ℘ is Weirstrass’ ℘-function.

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∂ We now proceed to show that any vector ﬁeld X(z) = g(z) ∂z with g an elliptic Ψ(z) ∂ function is a Newtonian vector ﬁeld X(z) = − Ψ (z) ∂z , and that we can in fact obtain an explicit formula for the representation of the analytic branched covering Ψ that characterises it.

Theorem 3.9 : Let f (z) be an elliptic function with fundamental periods 2w1 and 2w3 , let b1 , ..., br be the poles of f(z) in the fundamental period parallelogram. Suppose bk is of order βk , with principal part Gk =

Aβk k A1k + ··· + z − bk (z − bk )βk

(k = 1, ..., r).

(z) , and in fact there exist constants C and C such that Then f (z) = − ΨΨ(z)

Ψ(z) = C e−Cz

 r  k=1



 βk σ(z − bk )−A1k exp  (−1)j j=2

  Ajk ζ (j−2) (z − bk ) .  (j − 1)!

Proof : By Theorem 3.8 we know that r A2k ζ (z − bk ) + · · · A1k ζ(z − bk ) − f (z) = C + 1! k=1

βk −1

· · · + (−1)

Aβk k (βk −1) ζ (z − bk ) . (13) (βk − 1)!

Hence, r Ψ (z) A2k =C+ ζ (z − bk ) + · · · A1k ζ(z − bk ) − − (log(Ψ)) = − Ψ(z) 1!

k=1

· · · +(−1)βk −1 so that by integrating, and recalling that

∂ ∂z

Aβk k ζ (βk −1) (z − bk ) , (14) (βk − 1)!

log σ(z) = ζ(z), we obtain

r A2k ζ(z − bk ) + · · · A1k log σ(z − bk ) − − log Ψ(z) = C0 + Cz + 1 k=1

βk −1

· · · +(−1)

Aβk k (βk −2) ζ (z − bk ) , (15) (βk − 1)!

and by exponentiating and renaming C = e−C0 we have    βk r   Ajk A1k log σ(z − bk ) + ζ (j−2) (z − bk ) , (−1)j−1 Ψ(z) = C e−Cz exp −   (j − 1)! k=1

j=2

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12

and ﬁnally: Ψ(z) = C e−Cz

 r  k=1



 βk σ(z − bk )−A1k exp  (−1)j j=2

  Ajk ζ (j−2) (z − bk ) .  (j − 1)!

Corollary 3.10:

Let X(z) =

∂ g(z) ∂z

with g(z) an elliptic function, let f =

1 g,

∂ then X(z) is a Newtonian vector ﬁeld X(z) = − ΨΨ(z) (z) ∂z , with Ψ(z) as in Theorem 3.9.

4.

Future work

As mentioned in the introduction and throughout §3, the class of vector ﬁelds which are Newtonian vector ﬁelds, and is such that an explicit covering function can be constructed, is quite large and includes elliptic vector ﬁelds, rational vector ﬁelds, and some vector ﬁelds that have essential singularities. This, together with the visualisation method developed in [1], immediately shows that these families are all subject to visualisation without numerical integration. Having the explicit representation, and its geometric interpretation as a branched covering of the sphere onto itself, indicates that it might be possible to extend the works of [6, 7] and of [8], regarding the classiﬁcation of rational and elliptic Newton ﬂows to a larger class of Newton ﬂows. Moreover it seems plausible that some new results, in the vein of the work of [3, 4], on the dynamics for the ﬂows of the families of vector ﬁelds with explicit representation of their associated covering map, might be accessible from the explicit representation. On the other hand, the fact that elliptic vector ﬁelds are Newton vector ﬁelds, and since elliptic functions can be written in terms of theta functions, might provide some new results regarding theta functions, hence providing a link to number theory. Also in this direction, one can turn attention to the study of automorphic vector ﬁelds which could open up some further avenues of research. Work in these directions are underway and will be presented elsewhere.

Acknowledgments

The ﬁrst author would like to thank Professor Jes´ us Muci˜ no-Raymundo for introducing him to the study of complex vector ﬁelds and for many interesting, insightful and productive discussions. The other two authors wish to acknowledge and thank the ﬁrst author for the opportunity to work, under his guidance, on their undergraduate theses which led to some of the results of this paper. This work was made possible in part by UABC grant 0196.

References [1] Alvarez-Parrilla, A.: Complex analytic vector ﬁeld visualization without numerical integration, preprint, 2008. [2] Alvarez-Parrilla, A.; Muci˜ no-Raymundo, J.: Dynamics of Complex Analytic Vector Fields with Essential Singularities, preprint, 2008. [3] Benzinger, Harold E.: Plane autonomous systems with rational vector ﬁelds. Trans. Amer. Math. Soc. 326 (1991), no. 2, 465–483.

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[4] Benzinger, Harold E.: Julia sets and diﬀerential equations. Proc. Amer. Math. Soc. 117 (1993), no. 4, 939–946. [5] Burns, Scott A.; Palmore, Julian I.: The Newton transform: an operational method for constructing integrals of dynamical systems. Advances in ﬂuid turbulence (Los Alamos, NM, 1988). Phys. D 37 (1989), no. 1-3, 83–90. [6] Jongen, H.Th.; Jonker P. and Twilt, F.: The continuous, desingularized Newton method for meromorphic functions. Acta Appl. Math., 13, (1988), 81–121. [7] Jongen, H.Th.; Jonker P. and Twilt, F.: On the classication of plane graphs representing structurally stable rational Newton ﬂows. J. Comb. Th. Series B. 15(2), (1991), 256–270. [8] Helminck, G. F.; Kamphof, F. H.; Streng, M. and Twilt, F.: The Qualitative Behaviour of Newton Flows for Weirstrass’ ℘-Functions. Complex Variables and Elliptic Equations, 47:10, (2002), 867 - 880. [9] Hirsch, M.W.; Smale, S.: On algorithms for solving f (x) = 0. Comm. Pure Appl. Math., 32, (1979), 281–312. [10] Muci˜ no-Raymundo, J.: Complex structures adapted to smooth vector ﬁelds. Math. Ann. 322, (2002), 229–265. [11] Muci˜ no-Raymundo, J.; Valero-Vald´ez, C.: Bifurcations of meromorphic vector ﬁelds on the Riemann sphere. Ergod. Th. & Dynam. Sys. 15, (1995), 1211–1222. [12] Newton, T.; Lofaro T.: On using ﬂows to visualize functions of a complex variable. Mathematics Magazine, Vol. 69, No. 1 (Feb., 1996), 28–34. [13] Palmore, Julian I.; Burns, Scott A.; Benzinger, Harold E.: Ecology models and Newton vector ﬁelds. Nonlinearity in biology and medicine (Los Alamos, NM, 1987). Math. Biosci. 90 (1988), no. 1-2, 219–232. [14] Smale, S.: A convergent process of price adjustment and global Newton methods. J. Math. Econom. 3, (1976), 107–120. [15] Smale, S.: On the eﬃciency of algorithms of analysis. Bulletin of the Amer. Math. Soc., 13, 2, October 1985, 87–121. [16] Markusevich, A.I.: Theory of Functions of a Complex Variable, Vol. II. 1965, PrenticeHall.