On the separatrix graph of a rational vector field on the Riemann sphere
OOn the separatrix graph of a rational vector fieldon the Riemann sphere
Kealey Dias ∗ Bronx Community College of the City University of New York2155 University Ave. Bronx, New York, USA [email protected]
Antonio Garijo
Dept. d’Enginyeria Informàtica i Matemàtiques. Universitat Rovira i VirgiliAv. Països Catalans 26. Tarragona 43007, [email protected]
October 2, 2020
Abstract
We consider the rational flow ξ R ( z ) = R ( z )( d/dz ) where R is given by the quotient oftwo polynomials without common factors on the Riemann sphere. The separatrix graph Γ R is the boundary between trajectories with different properties. We characterize theproperties of a planar directed graph to be the separatrix graph of a rational vector fieldon the Riemann sphere. Keywords: vector fields, holomorphic foliations, separatrix graphMSC2010: 37C10, 34C05, 34M99, 37F75, 30F30
Complex differential equations have been playing an important role both in theoretical andin applied mathematics. Therefore, the study of these continuous dynamical systems couldbe interesting for wide areas of knowledge, from complex geometry to fluid dynamics. In thiswork we investigate some properties of a rational vector field defined in the Riemann sphere (cid:98) C . More precisely, we consider dzdt = R ( z ) , t ∈ R (1)defined by a rational map R : (cid:98) C → (cid:98) C given by R ( z ) = P ( z ) /Q ( z ) where P and Q aretwo polynomials that we assume without common factors. We also denote by n and m the ∗ The first author has been partially supported by the Association for Women in Mathematics Travel GrantOctober 2019 Cycle (NSF 1642548) and the second author has been partially supported by MINECO-AEIgrant MTM-2017-86795-C3-2-P. a r X i v : . [ m a t h . D S ] S e p egree of the polynomials P and Q , respectively. We assume that the degree of R , denoted by deg ( R ) = max { n, m } , is bigger or equal than . Equivalently, ξ R ( z ) = R ( z )( d/dz ) denotes therational vector field associated to R . We concentrate in rational vector fields on the Riemannsphere. For a deep investigation on other Riemann surfaces we refer to [APMnRSCYR20]and in the presence of an essential singularity to [APMnR17].These type of vector fields has been widely investigated from the local and global point ofview (see for instance [Háj66a, Háj66b, BT77, Sve79, NK94, Ben91, MnR02, GGJ07]) and wequote briefly some well known results of these systems. The fact that the vector field ξ R comesfrom a meromorphic map R provides specific properties to these kind of continuous dynamicalsystems. Among others ξ R does not present limit cycles or there are not center-focus problemsince the linear part of a critical point determines its behavior. More precisely, any simplezero α of P gives a sink/source or a center of ξ R according to Re ( P (cid:48) ( α )) is different or equalto zero, respectively. A multiple zero of P gives a critical point of ξ R with elliptic sectors.Finally, any root of Q gives a generalized saddle of the rational flow with hyperbolic sectors.A phase curve is called a separatrix if it tends to a saddle point. The separatrix graph , denotedby Γ R , of the rational flow ξ R is formed by the union of the closure of all the sepatrices of thesystem. The main goal of this work is to find a characterization of the separatrix graph of arational vector field.We mention two particular cases of rational vector fields. The first one is when the rationalmap is a polynomial P . The critical points of the polynomial flow ˙ z = P ( z ) are the zeros of P , and ∞ is a generalized saddle point. For more details of the classification of polynomialflows via a complete set of realizable topological and analytic invariants, we refer the reader to[BD10, Dia13]. Our approach will be based on the polynomial case, taking into account thatin the rational case has a richer structure in the phase space. One example of this differenceis that the separatrix graph of polynomial flow is always connected, which is not always thecase for rational flows.The second particular case of rational flows we highlight is the Newton’s flow when R ( z ) = − P ( z ) /P (cid:48) ( z ) where P is a polynomial. In this case, the critical points of Newton’s flow are sinks (zeros of P ) and simple saddles (zeros of P (cid:48) ) and ∞ is a source. Among other properties the separatrixgraph of a Newton’s flow is connected. In 1985 S. Smale proposed a question related to thecharacterization of the separatrix graph of a Newton’s flow ([Sma85]). In 1988 M. Shub,D. Tischler and R. Williams solved completely this question ([STW88]) characterizing theproperties of a planar graph to be the separatrix graph of a Newton’s flow.Based on the same question, our goal is to find a characterization of the separatrix graphof any rational vector field. An admissible graph is a planar graph with the main propertiesof the separatrix graph of a rational vector field (see §3 for a precise definition). The mainobjective of this paper is to prove the following result. Theorem A.
A planar directed graph Γ homeomorphic to the separatrix graph of a rationalvector field if and only if Γ is an admissible graph. Significant progress has been made towards solving this problem in the work of J. Muciño(see [MnR02]). That paper completely characterized the rational flows on compact Riemannsurfaces of genus g ≥ . The paper also gave a partial characterization of rational flows on (cid:98) C : those where the poles and equilibrium points are simple. It remains to show the completecase of rational flows on (cid:98) C , which is what we do in this paper.2n §2 we present the main properties of a rational flow, we define the separatrix graphof a rational vector field and deduce some of its properties. In §3 we introduce the abstractdefinition of an admissible graph, based on the properties of the separatrix graph of a rationalflow. Finally, in §4 we prove the characterization of a separatrix graph of a rational flowstated in Theorem A. In this section we recall the main aspects of the phase portrait of a rational flow (see [Háj66a,Háj66b, BT77, Sve79, NK94, Ben91, MnR02, GGJ07, BD10, Dia13]). We start studyingthe local dynamics of equation (1) in some punctured neighborhood of a critical point. Thelocal behavior of a rational flow around a critical point z (see Proposition 1) comes from itslocal normal form. For a simple zero of R the dynamics near z is conformally conjugate to ˙ z = R (cid:48) ( z ) z near the origin; for a multiple zero of R of order k ≥ one normal form is givenby ˙ z = z k / ( c + cz k − ) near the origin where c = Res(1 /R, z ) . Finally, the rational flow neara pole of R of order k ≥ is conformally conjugate to ˙ z = 1 /z k near the origin.In the next proposition we describe the local phase portrait, and we show the differenttypes of behaviors in Figure 1. Proposition 1.
Let R ( z ) be a holomorphic function in a punctured neigbourhood of z . (a) If z is a zero of R and R (cid:48) ( z ) (cid:54) = 0 , according to Re ( R (cid:48) ( z )) < , Re ( R (cid:48) ( z )) > or Re ( R (cid:48) ( z )) = 0 , then the phase portrait of (1) in a neigbourhood of z is a sink, a sourceor an isochronous center, respectively. In all the cases the index of ˙ z = R ( z ) at z is 1.(b) If z is a zero of R of order k ≥ , then the phase portrait of (1) in a neigbourhood of z contains the union of k − elliptic sectors, so the index of ˙ z = R ( z ) at z is k. (c) If z is a pole of R of order k ≥ , the phase portrait of (1) in a neigbourhood of z isa union of k + 1) hyperbolic sectors, and so the index of ˙ z = R ( z ) at z is − k. In Figure 1 we sketch the phase portrait of a rational flow in a neighborhood of criticalpoint. In the first row we show the local phase portrait of the simple zeros of R which couldbe sinks, sources, or isochronous centers. In the second row we show the phase portrait of amultiple zero of R with multiplicities k = 2 , . Finally, in the third row we show the localphase portrait of a pole of R with multiplicities k = 1 , .From Proposition 1 a multiple root of R is also called an elliptic critical point while a poleof R is called a saddle or a hyperbolic critical point.Given any non-critical point z in the Riemann sphere, we denote by γ ( t, z ) the solution ofthe Cauchy problem associated to (1) with initial condition γ (0 , z ) = z . Furthermore we canassume that γ is defined in its maximal interval of definition, thus γ ( · , z ) : ( t min , t max ) → (cid:98) C iscalled a trajectory through z . There are only two possibilities for the limit set of a trajectory:a critical point or a periodic orbit. In the first case critical points are sinks, sources orelliptic points (corresponding to t min = −∞ and/or t max = + ∞ ) or saddles (correspondingto t min > −∞ and/or t max < + ∞ ); and the second case corresponds to a periodic trajectory γ ( t + τ, z ) = γ ( t, z ) of minimal period τ > . The Poincaré-Bendixon Theorem of a rationalflow says that the ω − limit set of an orbit is either a critical point or a periodic orbit, andin the second case the nearby orbit must be periodic. The missing cases (limit cycles and aunion of singular points and phase curves) are not allowed due to the holomorphic structureof the flow. 3 ourcesink centerelliptic ( k = 3) elliptic ( k = 2) saddle ( k = 1) saddle ( k = 2) Figure 1:
Different local phase portraits near a critical point in a rational flow. In the first row asink, a source and a center; in the second row two examples of a critical point of elliptic type; andfinally in the third row two saddles.
We say that a trajectory γ is a separatrix if it lands in a (generalized) saddle point, orin other words, a separatrix is a trajectory for which the maximal interval of definition isdifferent from R . There are also different types of separatrices: an outgoing separatrix when t min > −∞ and t max = ∞ , an ingoing separatrix when t min = −∞ and t max < ∞ , an heteroclinic separatrix when t min > −∞ , t max < ∞ and lim t → t min γ ( t, z ) (cid:54) = lim t → t max γ ( t, z ) , and finally an homoclinic separatrix when t min > −∞ , t max < ∞ and lim t → t min γ ( t, z ) = lim t → t max γ ( t, z ) . Equivalently, we could define the separatrices as the union of all the stable and unstablemanifolds of all the hyperbolic points.There are several equivalent definitions of the separatrix graph of the vector field ξ R definedon the Riemann sphere (cid:98) C . We define the separatrix graph as the union of the closure of allthe separatrices, thus Γ R = (cid:91) γ separatrix γ. (2)We notice that every separatrix γ ( · , z ) : ( t min , t max ) → (cid:98) C is defined on an open interval,thus we denote by γ the separatrix γ and the two extremities lim t → t min γ ( t, z ) and lim t → t max γ ( t, z ) .4f γ is an outgoing/ingoing separatrix one the two extremities is a zero of R and the other isa pole of R , moreover when the separatrix is homoclinic/heteroclinic then both extremitiesare poles of R . Remark 1. If R is a polynomial of degree 2 using our definition of the separatrix graph (2) we obtain that Γ R = ∅ since there are not any separatrix trajectory. Moreover, the phaseportrait of R is well understood depending on the roots of R . Thus, ˙ z = R ( z ) is conformallyconjugate to ˙ z = i ( z − z − where ˆ C \ {± } is foliated by isochronous periodic orbits, orto ˙ z = ( z + 1)( z − where ˆ C \ {± } is foliated by orbits going from -1 to +1, or to ˙ z = z where ˆ C \ { } is foliated by orbits going from 0 to 0. From the above remark and hereafter we will assume that the degree of R is bigger or equalthan three and thus Γ R (cid:54) = ∅ . The separatrix graph Γ R is the boundary between trajectorieswith different properties. According the Markus Theorem ([Per01], Thm. 2, Sec. 3.11) theseparatrix graph Γ R is closed, so their complement (cid:98) C \ Γ R is open. Following [BD10, DES13]the connected components of (cid:98) C \ Γ R are called zones or canonical regions . Moreover, in everyzone all the trajectories are equivalent and could be classified into four types. See Figure 2. • A center zone C contains an equilibrium point z which is a center. From the topologicalpoint of view C \ { z } is a doubly connected region conformally isomorphic to D \ { } .Thus C \ { z } has infinite modulus mod ( C \ { z } ) = ∞ . Moreover, every center zone C is foliated by periodic orbits of the same period except the equilibrium point z . SeeFigure 2 first row. • An annular zone A is a doubly connected region with finite conformal modulus < mod ( A ) < + ∞ also foliated by periodic orbits of the same period. Each boundarycomponent of A is formed by one or several homoclinic/heteroclinic connections. Thevalue mod ( A ) is a conformal invariant of the double connected region A . However,deforming continuously the separatrix graph we can modify the value of mod ( A ) toany other value in (0 , + ∞ ) . See Figure 2 second row. • A parallel zone P is a simply connected region containing two different equilibriumpoints on the boundary of P , denoted by z and z , corresponding to the α − limit pointand ω − limit point for all trajectories in P, respectively. The boundary of P containsone or two incoming landing separatrices and one or two outgoing landing separatrices,and zero or more homoclinic/heteroclinic connections. See Figure 2 third row. • An elliptic zone E is a simply connected region containing exactly one equilibriumpoint z on the boundary, which is both the α − limit and ω − limit for all trajectories.The boundary of E consists of one incoming landing separatrix, one outgoing landingseparatrix, and zero or more homoclinic/heteroclinic connections. See Figure 2 fourthrow.In the case of a polynomial vector field only center, elliptic, and parallel zones exist([BD10]) since annular zones needs at least two poles of R and polynomial vector fields onlyhave a unique pole.In each zone we can define the rectifying coordinates that globally conjugate the vectorfield ξ R to the unit vector field d/dz ) . In any simply connected domain avoiding zeros of R , the differential dzR ( z ) has an antiderivative unique up to addition by a constant φ ( z ) = (cid:90) zr dwR ( w ) . w φw w w w w w w w C center zone half infinite cylinder w φw w w w w w w w w w w w w A annular zone finite cylinder z φw w z w w w w S stripparallel zone w w z w w w w φ + H half planeelliptic zone Figure 2:
The different types of zones: center, annular, parallel, and elliptic; and their correspondingimage under the rectifying coordinates φ : half infinite cylinder, finite cylinder, strip, and half plane.The points labeled w , . . . , w are poles of R (saddles) and z and z are zeroes of R (sinks, sources,centers or elliptic points). Note that φ ∗ ( ξ R ) = φ (cid:48) ( z ) R ( z ) ddz = ddz . The coordinates w = φ ( z ) are, for this reason, called rectifying coordinates . We will callthe images of zones under rectifying coordinates rectified zones . The rectified zones are of thefollowing types. See Figure 2. • The image of a center zone (minus a curve contained in the zone which joins the center z and a saddle equilibrium point in the boundary) under φ is a half infinite cylinder C . It could be either an upper half infinite cylinder or a lower half infinite cylinderdepending on the orientation of the closed trajectories in the center zone. In Figure 26first row) we show a center with the trajectories oriented counterclockwise and theircorresponding upper half infinite cylinder. • The image of an annulus zone (minus a curve contained in the zone which joint twosaddles equilibrium points each one in the two different boundaries) under φ is a finitecylinder A . It could be either an upper finite cylinder or lower finite cylinder dependingon the orientation of the trajectories in the annulus. In Figure 2 (second row) we show anannulus with the trajectories oriented counterclockwise and their corresponding upperfinite cylinder. • The image of a parallel zone under φ is a horizontal strip S . See Figure 2 (third row). • The image of an elliptic zone under φ is either an upper half plane + H or a lowerhalf plane − H . In Figure 2 (fourth row) we show an elliptic zone with the trajectoriesoriented counterclockwise and their corresponding upper half plane.We turn our attention to ∞ ; this point can be interpreted as the north pole of the Riemannsphere ˆ C . The usual approach to investigate the local phase portrait of (1) near ∞ is toconsider the change of variables w = 1 /z . Thus z = ∞ for (1) becomes w = 0 in the newvariable. The local behavior of ∞ mainly depends on the degrees of the polynomials P and Q . Using this approach the point of ∞ could be a critical or a regular point. We recall that z is a regular point if the rational flow (1) near z is conformally conjugated to ˙ z = 1 nearthe origin.We normalize (1) so that the point at infinity is always a regular point. We claim that ∞ is a regular point if and only if deg ( P ) = deg ( Q ) + 2 . To see the claim we assume that ∞ is a regular point or equivalently we assume that the index at ∞ is equal to 0. From thePoincaré Index Theorem ([Per01], Thm. 8, Sec. 3.12) we have that (cid:88) c I ( c ) = χ ( (cid:98) C ) = 2 where the sum is taken over the critical points of the rational function R ( z ) = P ( z ) /Q ( z ) .Assuming the ∞ is not a critical point and using Proposition 1 we have that (cid:80) c I ( c ) = n − m ,since every zero of R has index equal to their multiplicity as a root of P and every pole of R has index equal to minus their multiplicity as a root of Q . Concluding thus that when ∞ is not a critical point then n = m + 2 . In the next proposition we prove that we can alwaysassume that ∞ is a regular point. Proposition 2.
Any rational flow R ddz can be conformally conjugated by a Mobius transfor-mation to one with deg ( P ) = deg ( Q ) + 2 .Proof. Pick any regular point α for R not on a separatrix and such that P ( α ) Q ( α ) (cid:54) = 0 , wealso select a (cid:54) = 0 and let M ( z ) = azz − α be the Mobius transformation which sends α (cid:55)→ ∞ and ∞ (cid:55)→ a . Then ( M ) ∗ ( R ( z ) ddz ) = ˜ R ( w ) ddw , where ˜ R ( w ) = M (cid:48) ( M − ( w )) · R ( M − ( w ))= − aα (cid:0) αww − a − α (cid:1) · a n (cid:0) αww − a (cid:1) n + · · · + a b m (cid:0) αww − a (cid:1) m + · · · + b ( w − a ) m ( w − a ) n gives ˜ R ( w ) = − ( w − a ) aα · ( w − a ) m ( w − a ) n · a n ( αw ) n + a n − ( w − a ) ( αw ) n − + · · · + a ( w − a ) n b m ( αw ) m + b m − ( w − a ) ( αw ) m − + · · · + b ( w − a ) m = − ( w − a ) m +2 aα ( w − a ) n · P ( α ) w n + O ( w n − ) Q ( α ) w m + O ( w n − ) , where you can see that the degree of the numerator is n + m + 2 and the degree of thedenominator is n + m .Hereafter and without loss of generality we can assume that our rational flow given by (1)is such that deg ( P ) = deg ( Q ) + 2 . Remark 2.
Given P a polynomial we have that ˙ z = P ( z ) is conformally conjugate, via aMobius transformation, to ˙ z = ˜ P ( z ) / ( z − a ) n − where deg ( P ) = deg ( ˜ P ) . So, polynomial flowswrite as R = P/Q with deg ( P ) = deg ( Q ) + 2 and such that Q has a unique root. In a similarway, every Newton’s flow ˙ z = − P ( z ) /P (cid:48) ( z ) is conformally conjugate to ˙ z = ( z − a ) ˜ P ( z ) / ˜ Q ( z ) where deg ( ˜ P ) = deg ( P ) and deg ( ˜ Q ) = deg ( P ) − . In both cases the finite parameter a ∈ C plays the role of ∞ . As mentioned before, the Markus Theorem asserts that the separatrix graph and one orbitin each canonical region is enough to characterize the flow modulo topological equivalences.The separatrix graph alone is not enough; the next lemma exemplifies that different rationalflows can share the same separatrix graph.
Lemma 1.
The rational vector fields ˙ z = z / ( z − and ˙ z = i ( z − z − / ( z + 9) havean equivalent separatrix graph.Proof. We observe that in both cases ∞ is a regular point since deg ( P ) = deg ( Q ) + 2 . Wefirst consider the rational flow given by ˙ z = z z − and we denote by Γ its separatrix graph.The phase portrait of ˙ z = z z − exhibits an elliptic point located at the origin of multiplicity and a saddle point at z = 1 of multiplicity . Moreover, In both cases the local phaseportrait around each equilibrium point is given by four separatrices, two incoming and twooutcoming. We have that ˆ C \ Γ is formed by four elliptic zones since all the other possibilitiesare excluded. So, every separatrix trajectory connects the saddle point with the multiple pointand the complement is formed by four simply connected regions.We secondly consider the rational flow given by ˙ z = i ( z − z − z +9 and we denote by Γ itsseparatrix graph. Simple computations show that this rational vector field has four centerslocated at z = ± and z = ± and two simple saddle points at z = ± i . Moreover, everysimple saddle point has four separatrix trajectories, two incoming and two outcoming. Thus,the phase portrait has four center zones and the boundary of every center zone is formed bytwo heteroclinic connections. So, every separatrix trajectory connects the two simple saddles.In order to avoid this difficulty where different flows share the same separatrix graph, wecolor the vertices of the separatrix graph rather than including an orbit in each region. Moreprecisely, we distinguish between saddle points (white vertices) and the rest of the criticalpoints (black vertices). 8 Γ -3i 3i Γ Figure 3:
Sketch of the separatrix graph of the rational vector fields ˙ z = z / ( z − (left) and ˙ z = i ( z − z − / ( z + 9) (right). In the first case the complement of separatrix graph is formedby four elliptic zones (left) and in the second case by four center zones (right). This demonstratesthat the separatrix graph alone is not enough to determine topologically distinct phase portraits. In this section we will describe the conditions of an embedded planar directed graph to be theseparatrix graph of a rational vector field. Roughly speaking an admissible graph is a planarand directed graph that looks like a separatrix graph of (1).We first introduce some notion of graphs. We denote the planar and directed graph by
Γ = { V, E } , where V is the set of vertices and E the edges. In the set V we distinguishbetween two kind of vertices: white and black vertices. We will see later that white verticeswill correspond to saddle points of the vector fields while black vertices will correspond tosink, sources and elliptic points of the vector field.Given p ≥ , q ≥ and k ≥ we define the set of vertices and edges of Γ by, V = { b , · · · , b p , w , · · · , w q } E = { e , · · · , e k } . (3)where every oriented edge e i = ( x i , y i ) starts at the vertex x i ∈ V and finish at the vertex y i ∈ V , for ≤ i ≤ k ; an edge e = ( x, y ) is called a loop in the case that x = y . For everyvertex x we define the valence v ( x ) as the number of edges at the vertex x . In the case thatthe graph Γ exhibits a loop e = ( x, x ) , then this loop contributes 2 to the valence of the vertex x . In a similar way for every vertex x we define the cyclic reversals r ( x ) as the number ofreversals, from edges starting at x to edges finishing at x , when you made a turn around x .We notice that the valence counts the number of edges at the vertex, while the cyclic reversalonly counts the number of incoming and outgoing edges at the vertex. See Figure 4. valence = 5reversals=2valence = 2reversals = 2valence=4reversals = 4 valence= 9reversals= 4 Figure 4:
Four examples counting the valence and reversals of a concrete vertex. z is asaddle point of order k ≥ then v ( z ) = r ( z ) = 2( k + 1) , if z is a multiple root of R of order k ≥ then r ( z ) = 2( k − and finally if z is a source or a sink of R then r ( z ) = 0 . Wenotice that a priori we do not know what is the valence of a concrete root (simple or multiple)of R from its local behavior.We also introduce the following quantities associated to the graph Γ . We denote by V = q (cid:88) i =1 v ( w i ) the total valence at the white vertices and R = p (cid:88) i =1 r ( b i ) the total cyclicreversals at the black vertices.We can define an admissible graph Γ as a planar and directed graph with the main prop-erties of the separatrix graph of a rational vector field. We notice that the separatrix graph ofa rational vector fields involves three main ingredients. The first one is the local behavior atthe vertices, the second one is the Poincaré index formula and finally what kind of domainsare allowed at the complement of Γ . Definition 1.
Let p ≥ , q ≥ and k ≥ three natural numbers. We consider Γ a planarand directed graph Γ = { V, E } where V = { b , · · · , b p , w , · · · , w q } are the vertices and E = { e , · · · , e k } are the edges. We say that Γ is an admissible graph if and only if the followingconditions are satisfied(a) There are not isolated vertices.(b) Every edge is incoming or outgoing from a white vertex.(c) Every white vertex w verifies that v ( w ) = r ( w ) is an even number greater than or equalto 4.(d) Every black vertex b verifies that r ( b ) is an even number greater than or equal to 0.(e) p + R ≤ − q + V (Poincaré index formula)(f ) The complement of Γ is formed by simply connected and doubly connected regions. If N + 1 is the number of connected components of Γ , then C \ Γ has N annular regions.Every annular region is doubly connected whose boundary is formed by white vertices,and both boundary components have the same orientation. See figure 6.(g) There are c := V − q + 2 − (cid:0) p + R (cid:1) ≥ center regions. Every center region is simplyconnected and has boundary which is an oriented cycle formed by white vertices. Seefigure 5.(h) There are R elliptic regions. Every elliptic region is simply connected and whose bound-ary is a cycle of exactly one black vertex and one or several white vertices, oriented fromthe unique black vertex to itself. See figure 7.(i) The rest of components of C \ Γ are formed by parallel regions. Every parallel regionis simply connected whose boundary contains two black vertices and two oriented paths(not necessarily disjoint) both going from one black vertex to the other black vertex. Seefigure 8. Lemma 2.
Let Γ be the separatrix graph of a rational vector field (1) , then Γ is an admissiblegraph. roof. We denote by Γ the separatrix graph of ˙ z = R ( z ) . We color the saddle points whiteand sinks, sources, and multiple points black. We assume that the rational flow given by R writes as P ( z ) /Q ( z ) , with deg ( P ) = n and deg ( Q ) = n − (see Proposition 2) and n ≥ (seeRemark 1). Since deg ( Q ) ≥ , then we have at least one white vertex, q ≥ , and at least twoedges, k ≥ . The first four properties of the definition of an admissible graph are triviallysatisfied from the local behavior at critical points (see Proposition 1).We derive property (e) from Poincaré index formula. Firstly, we deal with saddle points(white vertices). By assumption every saddle point corresponds to a root of the polynomial Q . Writing Q ( z ) = q (cid:89) i =1 ( z − w i ) d i , we have that V = q (cid:88) i =1 v ( w i ) = q (cid:88) i =1 d i + 1) = 2( n − q ) , (4)since a root w i of multiplicity d i has a valence equal to d i + 1) (see Proposition 1) and deg ( Q ) = n − .Secondly, we deal with sinks, sources and elliptic points (black vertices) which are roots of P . However, in that case we need to take into account center points (if there are any) sincethey are roots of P but not vertices of Γ . The number of sinks, sources and elliptic pointsverify p (cid:88) i =1 (cid:18) r ( b i )2 + 1 (cid:19) = R p ≤ n, (5)since they are roots of P and deg( P ) = n . Furthermore, we have that the number of centerzones is exactly n − R − p .Finally, combining Equations (4) and (5) we obtain the desired inequality p + R ≤ − q + V . The rest of the properties follow from the four types of zones discussed in §2. We onlycheck the number of center regions (g). From Equation (5) the number of center is equal to n − (cid:0) p + R (cid:1) . Replacing n from Equation (4) we obtain that the number of center zones isexactly c := V − q + 2 − (cid:18) p + R (cid:19) . The goal of this section is to prove Theorem A which states which planar graphs correspondto the separatrix graph of a rational flow. More precisely, Theorem A states that a planar anddirected graph Γ corresponds to the separatrix graph of a rational vector field if and only if itis an admissible graph. The precise definition of an admissible graph is given in §3. Almosttrivially, any rational vector field must have separatrix graph satisfying these conditions, sincethe conditions were designed with the rational separatrix graph in mind (See Lemma 2).11ow we will show that the admissibility conditions are enough. The steps of the proof arelisted here, and proving each step will follow.1. Build rectified zones from Γ , and glue these together to create a rectified surface M ∗ with punctures. See §4.1.2. Construct an atlas for M ∗ to show that it is a Riemann surface, and use the chartsaround the punctures to define the closure M . See §4.2.3. Use the Euler characteristic to show that M is homeomorphic to ˆ C , and the Uniformiza-tion Theorem then gives that it is, in fact, isomorphic to ˆ C . See §4.3.4. Endow M ∗ \ { w , . . . , w q } with the vector field ddz in the natural charts, and extendthis holomorphically to the vector field ξ M , holomorphic on M \ { w , . . . , w q } andmeromorphic on M . See §4.4.5. With all these in our hands we can prove Theorem A. We show that the induced vectorfield must be a rational vector field of the form (1) with deg( P ) = deg( Q ) + 2 . See §4.5. M ∗ with punctures Let
Γ = { V, E } be an admissible graph with q white vertices w , . . . , w q , p black vertices b , . . . , b q ; and k edges denoted by e , . . . , e k (see §3 for details).We will construct the four different types of rectified zones as a building blocks of therectified surface M ∗ : half-infinite cylinders C , finite cylinders A , half-planes ± H , and strips S . We will first explain their topology, and afterwards explain their geometry, since it turnsout that choosing the lengths of the edges joining white vertices on the boundaries is not asstraightforward as one would initially imagine.For each simply connected component of ˆ C \ Γ which is bounded by a counterclockwise(resp. clockwise) oriented cycle with N edges connecting M white vertices, construct anupper (resp. lower) half-infinite cylinder C = H ± /L Z , where the oriented cycle is on R /L Z and L is the sum of the N segment lengths between the white vertices to be determined (see§4.1.2). Thus given any L > the above construction of the half-infinite cylinder C dependson L . See Figure 5. w w w w w w w w w w half-infinite cylinder C Figure 5: Example of a simply connected region bounded by a clockwise oriented cycleformed by four white vertices and five edges (left), and the rectified zone C a half-infinitecylinder (right). 12or each doubly-connected region, construct a finite cylinder A as follows (see Figure6). Take the oriented boundary component which is an oriented cycle containing N edges w w w w w w w w w w w w w w A finite cylinderFigure 6: Example of a doubly connected component bounded by two clockwise orientedcycles, one formed by two white vertices and two edges and the other formed by four verticesand five edges (left), and a corresponding rectified zone A a finite cylinder (right).connecting M white vertices where the annulus is to the left of the boundary component andsend it to R /L Z , where L is the sum of the N segment lengths between the white vertices tobe determined (see §4.1.2). Take the oriented boundary component where the annulus is tothe right and send it to R /L Z + i , where L is the same as for the other boundary component.This is a finite cylinder with height 1 and circumference L . As before this construction couldbe done for every L > . As we mention before, we can deform continuously the separatrixgraph such that mod ( A ) = L . Note that there is a choice of shear, the relative positions ofthe white vertices on the upper and lower boundaries. b w w w w w w H + half-planeFigure 7: Example of simply connected component with exactly one black vertex on theboundary and three white vertices forming a counterclockwise cycle with four edges (left),and their corresponding rectified zone H + a half-plane (right).For each simply connected component with boundary oriented from the unique blackvertex to itself, map this boundary to the real line such that the orientation of the boundaryis from −∞ to + ∞ and such that ±∞ correspond to the black vertex. Take the upperhalf-plane H + (resp. lower half-plane H − ) if the component is to the left (resp. right) of itsboundary.For each simply connected component with exactly two black vertices on the boundary, theboundary consists of two sets of edges (not necessarily disjoint) which connect one black vertexto the other, respecting the orientation of the edges (see Figure 8). Construct a horizontal13 b w w w w w w w w S strip w w Figure 8: Example of simply connected component with exactly two black vertices and fourwhite vertices on the boundary, the boundary consists of two sets of edges which connectone black vertex to the other. More precisely, b → w → w → w → w → b and b → w → w → b (left); and the corresponding rectified zone S a strip (right).infinite strip with height 1 such that the upper and lower boundaries are horizontal lines, eachcorresponding to the two boundary sets of edges. There is again a choice of relative positionsof the white vertices on the upper and lower boundaries. We now discuss how to set the lengths of the heteroclinic and homoclinic segments on theboundaries of these regions. It would be simplest to set all of these lengths equal to 1, butthis cannot (usually) be done, as will now be explained. Each annular region (if they areany) has two boundary components, say one which has N edges and the other has N edges.The lengths of these boundary components need to be equal, since the trajectories in annularregions are isochronous. Since in general N (cid:54) = N , setting each edge length equal to 1 wouldgive one boundary component length N and the other boundary component length N . Eventhough the lengths cannot usually all be set to equal 1, the following result shows that theredoes exist some consistent assignment of lengths for every admissible graph. Proposition 3.
For every admissible graph, let x , . . . , x n be the lengths of the edges that con-nect white vertices to white vertices. There exists assignment of positive numbers to x , . . . , x n such that for each of the N annuli, the lengths of each of the two boundary components areequal. The proof, relegated to the Appendix, will apply a result by Dines [Din27] regarding theexistence of solutions to systems of linear equations which have all positive components. M ∗ Utilizing admissible Γ and Proposition 3, we construct half-infinite cylinders C i , finite cylinders A i , half-planes ± H , and strips S i as subsets in C . We define M ∗ = ( C i (cid:116) A i (cid:116) ± H (cid:116) S i ) / ∼ , (6)where ∼ is the equivalence relation such that all points corresponding to the same whitevertex are identified, and each pair of points on the two occurrences of any separatrix (edgeof Γ ) are identified by isometry. 14e will make charts of the neighborhoods of the boundary components of M ∗ to showthat each corresponds to one point. The natural ( p + c ) -point closure of M ∗ is denoted M and is called the rectified surface . We notice that these p + c points correspond to the p blackpoints and the c centers (see Definition 1 (g)) that are omitted in the construction of M ∗ .The construction of the Riemann surface structure on M is contained in the constructionbelow of an atlas. M M We show that M is a Riemann surface by constructing an atlas ( U i , η i ) for M with holomor-phic transition maps. There are obvious charts over the interior of each half-infinite cylinders C i , finite cylinders A i , half-planes ± H , and strips S i , and the transition maps are, at worst,translations in C . It remains to show charts over: • points on edges of Γ (separatrices), • the white vertices { w i | i = 1 , . . . , q } , and • the punctures of M ∗ .We treat each case in turn. Firstly, for points on edges we note the following. Since eachedge is on the upper boundary of exactly one rectified zone and on the lower boundary ofexactly one rectified zone, we define a neighborhood of a point on a edge in the natural way:construct sufficiently small half-disks of the same radius in the upper and lower zone, andidentify by isometry (see Figure 9). z D z D Figure 9: Sketch of the construction of a neighborhood of a point z in an edge of the admissiblegraph.Secondly, let w := w i be one of the q white vertices of Γ . We know that the valence at w is equal to m + 1) with m ≥ , since Γ is an admissible graph (see Definition 1 (c)).Thus, w is on the boundary of m + 1) rectified zones: m + 1 on lower boundaries of half-planes, strips, or (finite or half-infinite) cylinders, and m + 1 copies on upper boundaries. Wedefine a chart around each w as follows. Let D ± k , k = 1 , . . . , m + 1 be the upper or lowersemi-disk with center w in either a strip, half-plane, half-infinite cylinder, or finite cylinder, + if an upper semi-disk and − if a lower semi-disk, and with radius r sufficiently small. Theneighborhood of w is then the half-disks taken with proper identification, { D ± k } / ∼ , since thismaps univalently to a sufficiently small open disk in C by an ( m + 1) -st root. In Figure 1015 D +1 w D − wD − w D +2 z (cid:55)→ ( z + w ) D +1 D − D +2 D − Figure 10: We sketch the construction of the neighborhood of a white vertex w with valence 4,which corresponds to m = 1 . Thus w is on the boundary of four rectified zones: an upper halfplane (top left), a strip (bottom left), a lower half plane (top right) and an upper half-infinitecylinder (bottom right). Suitable identification of these half-disks gives a neighborhood of w in M which maps univalently to a disk in C under a suitable square root.we sketch the situation where m = 1 which maps to a small disk in C under a suitable squareroot.Thirdly, we now construct charts in neighborhoods U ⊂ M covering each boundary com-ponent of M ∗ . The boundary components of M ∗ come from the black vertices of Γ or theends of half-infinite cylinders. The latter case is simpler, so we begin there.Each half-infinite cylinder C is conformally isomorphic to D \ { } ⊂ C . This chart extendshomeomorphically to the closure, by mapping the puncture ∞ [ C ] to ∈ C . We rememberthat there are c cylinders and hence c such punctures (see Definition 1 (g)).Next, consider the boundary components of M ∗ which correspond to black vertices in Γ { b i | i = 1 , . . . , p } which only have edges directed toward (resp. away from) the black vertex.Let b := b i be one of the p black vertices of Γ with this property. Each simply connectedcomponent with those edges on their boundaries must be a strip zone. Gluing those strips bythe corresponding separatrices and truncating such that (cid:60) ( z ) > R (resp. (cid:60) ( z ) < R ) showsthat a neighborhood U [ b ] of b is a half-infinite cylinder to the right (resp. left). The conformalisomorphism z (cid:55)→ exp (cid:16) ∓ πiρ z (cid:17) maps this cylinder to a punctured neighborhood of ∈ C ,where ρ ∈ H is the number that gives the height and shear of the identification. This localchart extends homeomorphically to its closure (see Figure 11).The only case remaining is for a black vertex b := b i which has adjacent edges directedboth towards and away from it. There is a neighborhood U [ b ] of the boundary at infinity inrectifying coordinates that we will show is doubly connected and has infinite modulus. Thus,showing the boundary is a single point. Lemma 3.
All simple, closed curves in U [ b ] are either homotopic to the bounded boundarycomponent γ or homotopic to a point.Proof. If γ is not homotopic to the boundary component γ , then there exists a simple curve γ which joins γ to infinity, where γ and γ do not intersect (see Figure 12). The set U [ b ] \ γ is simply connected. Therefore, γ is homotopic to a point since it is contained entirely in asimply connected set.By Lemma 3, U [ b ] is doubly connected since every non-trivial, simple closed curve in U [ b ] ishomotopic to γ . We now use Grötzsch’s inequality to show that this doubly connectedregion has infinite modulus. Indeed, it is easy to see in rectifying coordinates that one16 [ b ] cylinder ρ ∈ H (cid:60) ( z ) = R Figure 11: A neighborhood U [ b ] of b := b i , where b i ⊂ Γ is a black vertex with four edges in Γ directed away from it. Gluing those strips by the corresponding separatrices and truncatinggives that U [ b ] is a half-infinite cylinder to the left (the red dashed lines in the figure areidentified by ρ ∈ H ). Therefore, U [ b ] is conformally isomorphic to D \ { } ⊂ C , and the chartextends homeomorphically to the puncture.can construct infinitely many disjoint annuli contained in U [ b ] and homotopic to γ whichhave modulus bounded away from zero since the H and strips are unbounded. Therefore, mod (cid:0) U [ b ] (cid:1) = ∞ since Grötzsch’s inequality gives mod (cid:0) U [ b ] (cid:1) ≥ (cid:80) ∞ i =1 mod ( A i ) , where thelatter must be infinite since each modulus mod ( A i ) is bounded away from zero. Therefore, U [ b ] is conformally isomorphic to D ∗ and can be extended homeomorphically by mapping thepuncture ∞ [ b ] to ∈ C .Once we have defined a neighborhood U for every point in M , we notice that all transitionmaps are translations or compositions of translations with conformal isomorphisms. There-fore, we have made an atlas with holomorphic transition maps on the Hausdorff space M ,and we can conclude that M is a Riemann surface. M isomorphic to (cid:98) C In this section we prove the following result.
Proposition 4. M is homeomorphic to (cid:98) C . We first need a lemma regarding the Euler characteristic.
Lemma 4.
An embedded planar graph with N + 1 connected components separated by N annular regions satisfies v − e + f = 2 , where f is the number of simply connected faces(including the face containing ∞ ).Proof. It is well known that v − e + f = 2 holds for connected graphs embedded in the planewith v vertices, e edges, and f faces which are simply connected. If there are additionally N annular regions, one may introduce one edge per annulus that connects a vertex on one bound-ary component of the annulus to the other boundary component. This yields a connectedembedded planar graph which has v vertices, e + N edges, and f + N faces. Substituting inthe equation gives the same result. 17 γ γ U [ b ] Figure 12: If γ is not homotopic to the boundary component γ , then there exists a simplecurve γ which joins γ to infinity, where γ and the γ do not intersect. The curve γ mustbe homotopic to a point since it is entirely contained in the simply connected set U [ b ] \ γ . Proof of Proposition 4.
We utilize the Euler characteristic. Note that M has a topologyinduced by the topology of C . The identifications in constructing M give a graph on M withthe same topology as Γ ⊂ C . By Lemma 4, Γ satisfies v − e + f = 2 , where f counts theexterior face but not the annular regions. Hence, M is homeomorphic to (cid:98) C . Corollary 1. M is isomorphic to (cid:98) C .Proof. Since M is a simply connected Riemann surface that is compact, it must be confor-mally equivalent to the Riemann sphere (cid:98) C by The Uniformization Theorem. ξ M We will show in this section that the singularities of ξ M are equilibrium points and poles.The vector field ξ M is defined on the half-infinite cylinders C , finite cylinders A , half-planes ± H , strips S , and neighborhoods of separatrix points as ( η ) ∗ ( ξ M ) = ddz . It remains to showwhat the extension of the vector field is at the punctures and at the white vertices.Recall from §4.2.1 that a neighborhood U [ b ] of a puncture ∞ [ b ] of ξ ∗M that corresponds toa black vertex b of Γ with incoming only or outgoing only edges corresponds to a truncatedunion of strips identified on the boundary to form a cylinder (recall also Figure 11). There isa chart η : U [ b ] → V , where V is a neighborhood of , such that ( η ) ∗ ( ξ M ) = ( ϕ ) ∗ (cid:18) ddz (cid:19) = ∓ πiρ z · ddz , (7)where ϕ ( z ) = exp (cid:16) ∓ πiρ z (cid:17) . The extension of the chart at the puncture ∞ [ b ] leads to aholomorphic extension of ξ M at the puncture by ξ M = 0 .18 neighborhood U [ C ] of a puncture ∞ [ C ] stemming from a cylinder zone is nearly identicalto the above. The total length L of homoclinics on the boundary gives the map ϕ ( z ) =exp (cid:0) ∓ πiL z (cid:1) giving a vector field which extends to ξ M = 0 at the puncture.A neighborhood U [ w ] ⊂ ξ ∗M of a white vertex w := w i with valence v := v i correspondsto a v/ cover of a sufficiently small neighborhood of . The induced vector field can becalculated: ( η ) ∗ ( ξ M ) = (cid:0) z /v (cid:1) ∗ (cid:18) ddz (cid:19) = 2 v z − ( v/ − ddz (8)in another sufficiently small neighborhood of since the v/ covering has constant vector field ddz . Therefore, the vector field ξ M has a pole of order v/ − at w .A neighborhood U [ b ] ⊂ ξ ∗M of a puncture ∞ [ b ] of ξ ∗M that corresponds to a black vertex b of Γ with both incoming and outgoing edges corresponds to a restricted union of half-planesand strips. We wish to show that the holomorphic extension of η for the chart η [ b ] : U [ b ] → V ∗ to ∈ V gives that the extended vector field to the puncture ∞ [ b ] is a zero of order m = r/ , where r := r ( b ) is the number of cyclic reversals (see §3) of the edge orientationwhen traversing around b . We do this by an index argument. More specifically we will showthat the index of a sufficiently small curve around 0 is m . Consider the (piecewise smooth)Jordan curve about ∞ [ b ] in U [ b ] that corresponds to the curve γ [ b ] in rectifying coordinates withhalf-circles in each upper and lower half-plane and appropriate line segments in the half-stripswith clockwise orientation so that ∞ [ b ] is to the left of the curve (see Figure 13). The curve U [ b ] γ [ b ] γ −→ η Figure 13: A neighborhood U [ b ] ⊂ ξ ∗M of a puncture ∞ [ b ] of ξ ∗M for a black vertex b of Γ with both incoming and outgoing edges corresponds to a restricted union of half-planes andstrips (left). The vector field in these regions is ddz (black arrows). The neighborhood of zero V ∗ = η ( U [ b ] ) containing the simple closed curve γ = η ( γ [ b ] ) (right). The pink arrows are thetangent vectors to γ with a counterclockwise orientation, and the black arrows are the vectorsof ( η ) ∗ ( ξ M ) along γ . γ [ b ] maps to another Jordan curve γ in V ∗ under η since η is univalent. Moreover, ( η ) ∗ ( ξ M ) isnever 0 along γ . The angles between ( η ) ∗ ( ξ M ) on γ and the tangent vectors of γ are the sameas the angles between the vector field ddz and the tangent vectors of γ since η is conformal(see Figure 13).This implies that on γ , ( η ) ∗ ( ξ M ) will be tangent to γ exactly r times, and alternatingwith pointing along the orientation of γ and against the orientation of γ when we travelalong γ . Along the orientation of γ , on the arcs between the tangents going along to the19angents going against, the vector field must be pointing inward, since this corresponds towhat happens in rectifying coordinates. This implies that the vector field must be rotating inthe same orientation as γ , otherwise there would be a place in this arc where there is anothertangent. So with respect to the tangents of γ , ( η ) ∗ ( ξ M ) has rotated r/ times. We must addone more time around, accounting for the index of γ . This gives that ( η ) ∗ ( ξ M ) has index r/ at 0. We now show that there is a conformal isomorphism
Φ :
M → (cid:98) C that induces a vector field ξ R = R ( z ) ddz = P ( z ) Q ( z ) ddz with deg( P ) = deg( Q ) + 2 having separatrix graph homeomorphic to Γ . Proof.
By the Uniformization Theorem, there exists an isomorphism
Ψ :
M → (cid:98) C , whichis unique up to post composition by a Möbius transformation and induces a vector field Ψ ∗ ( ξ M ) defined on (cid:98) C . Choose Ψ such that Ψ ( a ) = ∞ , where a is a regular point that is noton a separatrix. Then Ψ ∗ ( ξ M ) is a meromorphic vector field on (cid:98) C , expressed in canonicalcoordinates as g ddz . Since g is meromorphic on (cid:98) C , it is rational; and we will show that it isof the form P ( z ) Q ( z ) ddz with deg( P ) = deg( Q ) + 2 . We know by §4.4 that g has q poles of order m i at Ψ( w i ) , so the degree of Q is (cid:80) si =1 m i + 1) =: n . We will show that the degree of thenumerator P is n + 2 .Recall r ( b i ) is the number of edge orientation cyclic reversals at b i , and let d ( b i ) := v ( b i ) − r ( b i ) be the number of non-reversals in edge orientation at b i . The degree of the numeratorof R is the sum of equilibrium points, counting multiplicity deg( P ) = c + p (cid:88) i =1 ( r ( b i ) / c + p + 12 p (cid:88) i =1 r ( b i ) . (9)From Lemma 4, the embedded Γ satisfies v − e + f = 2 . The number of black and white verticesgives v = q + p . The number of edges is e = (cid:80) qi =1 m i + 1) − h , the number of separatrixdirections, minus the number of homoclinic or heteroclinic. This further simplifies to e =2 n + 2 q − h by using n := (cid:80) si =1 m i + 1) . The number of faces is f = c + (cid:93) half-planes + (cid:93) strips.The number of half-planes is (cid:80) pi =1 r ( b i ) , and the number of strips is (cid:80) pi =1 d ( b i ) . This gives v − e + f = 2 q + p − (2 n + 2 q − h ) + c + p (cid:88) i =1 r ( b i ) + 12 r (cid:88) i =1 d ( b i ) = 2 . (10)Combining this with Equation (9) gives deg( P ) =2 − q + 2 n + 2 q − h − p (cid:88) i =1 r ( b i ) − p (cid:88) i =1 d ( b i )=2 + 2 n + q − h − p (cid:88) i =1 ( r ( b i ) + d ( b i )) . (11)Now (cid:80) pi =1 ( r ( b i )+ d ( b i )) is the number of landing separatrices, which equals (cid:80) qi =1 m i +1) − h .20ombining with Equation (11) gives deg( P ) =2 + 2 n + q − h − ( q (cid:88) i =1 ( m i + 1) − h )=2 + 2 n + q − h − ( n + q − h )= n + 2 . (12) Appendix
Here we prove Proposition 3.
Proof of Proposition 3.
Without loss of generality, we may assume that x , . . . , x n are thelengths of the edges that are on the boundary of at least one of the N annular regions,since all other edges have lengths which may be chosen freely. Each annular region has twoboundary components: one on the left of the annulus, respecting graph orientation, and oneon the right. The N annular regions give N homogeneous linear equations in n variables.Each equation in the system can be written as (cid:88) i ∈ I (cid:96) x i = (cid:88) i ∈ I r x i , (13)where I (cid:96) is the subset of indices corresponding to the left-hand boundary component of thatannulus, and I r is the subset of indices corresponding to the right-hand boundary componentof the same annulus. This system of equations written in the form (13) has the followingproperties:(a) There is at least one x i on each side of each equation.(b) Each x i can appear at most twice in the system. If it does appear twice, it appears onceon the left and once on the right of two different equations.(c) If two equations have the same x i on one side, then their other sides must not have any x j in common.We explain the above properties. Property (a) is necessary since each annulus has at leastone edge on each boundary component. Property (b) is due to the fact that each edge canbe on the boundary of at most two annuli; and if it is on the boundary of two annuli, it is onthe left-hand boundary of one annulus and the right-hand boundary of the other. Property(c) reflects that if two annuli share an edge on one boundary component, then their otherboundary components share no edges. Indeed, the core curves in the annuli separate thesesecond boundary components.In Figure 14 we show the two different configurations of two annuli sharing some edges,where we can see the above three properties into a concrete example. On the left hand sideof this figure we show the not nested case where both annuli share the edge x . Using thenotation introduced before, we have the following two equations (cid:26) x + x = x + x + x + x x + x + x = x + x x x x x x x x x x x x x x x x x x x x Figure 14: Examples of two annuli sharing some edges. On the left hand side the two annuliare not nested and they share the edge x , on the right hand side the two annuli are nestedand they share the edges x , x and x .On the right hand side of the figure we show the nested configuration where both annulishare the edges x , x and x . In this concrete example the two linear equations writes as (cid:26) x + x = x + x + x x + x + x + x + x + x = x + x . Applying a result by Dines [Din27] will show that there exists a solution to this systemhaving all positive components, henceforth called positive solutions .The method of Dines is an inductive method to reduce the system of equations to asystem having one equation less, where the initial system admits a positive solution if andonly if the new system does. A requirement to use this result is that in no step of theinduction do any of the equations have all positive or all negative coefficients when writtenin standard form. Clearly, such an equation cannot have a positive solution. We will showthat at each inductive step in our systems (13) that the equations will never have all positiveor all negative coefficients; equivalently, at least one term on the left and on the right havepositive coefficients. We will first review the procedure of Dines and then apply it to our case.Consider a system of N homogeneous linear equations of the form n (cid:88) i =1 a ri x i = 0 , r = 1 , . . . , N. (14)The procedure of Dines first partitions the indices of the first equation according to the signsof the coefficients: let I be the set of i such that a i ≥ and let J be the set of j such that a j < . Then the first equation is rewritten as (cid:88) i ∈ I a i x i = − (cid:88) j ∈ J a j x j , (15)and the remaining N − equations are each written in the form (cid:88) j ∈ J a rj x j = − (cid:88) i ∈ I a ri x i , r = 2 , . . . , N, (16)22here it is understood that the i and j are from the same partition of indices as determined bythe first equation. Next, the first equation is multiplied by each of the other N − equationsto create a new system of N − linear equations in the new variables x ij := x i x j : (cid:88) i a i x i · (cid:88) j a rj x j = (cid:88) j a j x j · (cid:88) i a ri x i . (17)The procedure is repeated until there is only one remaining equation in y i := x i x i · · · x i N , i j = 1 , . . . , n for all j n · N (cid:88) i =1 b i y i = 0 , (18)which has a positive solution y i = − (cid:88) j b j (19) y j = (cid:88) i b i , (20)where again the indices i and j are partitioned such that b i ≥ and b j < . Dines provesthat the existence of this positive solution guarantees the existence of a positive solution forthe original system, as long as at each step of the iteration, no equation has only positive oronly negative coefficients (in standard form).Let us apply this procedure to our case. We will need to show that at each step of the pro-cedure, all equations in the system have both positive and negative coefficients (equivalently,at least one term on each side with positive coefficient). It will be enough to show that eachintermediate step satisfies the three properties above. Applying to the annulus problem, theequations (15) and (16) take the form (cid:88) { i }⊆ I x i = (cid:88) j ∈ J x j , r = 1 (21)and (cid:88) { j }⊆ J x j = (cid:88) { i }⊆ I ± x i , r = 2 , . . . , N, (22)since all coefficients are , or − . Note that both sides of the first equation have onlypositive coefficients as written. The r th equation as presented has left hand side which hasonly non-negative coefficient terms because of property (b). If the left is non-zero, the righthand side has at least one positive coefficient term by property (a); if the left equals zero, theright hand side has at least one positive and one negative term also by property (a). Sinceboth sides of (21) have positive coefficients, the signs of the product coefficients of (21) and(22) is determined by the signs of the coefficients of (22). Hence, multiplying the left sidesof (21) and (22) gives only positive coefficient terms or zero on the left, and as before, atleast one positive coefficient term on right in the first case and at least one positive and onenegative in the second case.first equation (+) = (+) (23)rth equation (+) = (+) + (+ or − ) or − ) + (+ or − ) (24)Therefore the equations in the resulting system have not all positive or all negative coefficientsin standard form (Property (a)). There is, however, a detail that needs to be considered: it23s a priori possible that some of the terms on the left may be equal to some on the right inthe product, and simplifying could hypothetically cause the equation to only have positiveor negative coefficient terms in standard form. Let us see that this cannot happen with oursystem. You will get cancelling terms if the product of (21) and (22) has some x i j both onthe right and on the left. The i will come from the left side of (21) and the j would have tobe from the left side of (22). Now the indices j on the left of (22) are some subset of the j on right of (21). The right of (22) gets multiplied by all of these j , so you would only get thesame pair i j if the right side of equation (22) contains some i that is on left of (21). Thiscannot happen by property (c) on (21) and (22).Now we need to show that the new system of equations in x ij satisfies the same threeproperties so that we can iterate this argument. First note that the coefficients of the productsystem are , , and − . The new system inherits property (a) by the paragraph above. Wenow show that it also inherits property (b) That is, every x ij can appear at most twice inthe N − equations, and if it appears twice they are in different equations with oppositesign in standard form (or each positive on opposite sides). By the argument in the precedingparagraph, no x i j can be both on the left and right of the same equation. The variable x i j can not appear anywhere else on the left in the product system (17), since the j comes fromthe left of (22), and this cannot appear anywhere else on the left by property (b) since it ison the right of (21). The same x i j can appear at most once on the right since i alreadyappears on left in (21) so can appear at most once on right of the system (and both havepositive coefficients.)It now remains to show that if two equations in the product have an x i j once on eacha respective side, that the opposite sides of the same equations share no other x i j . If theproduct gives a common x i j , it must stem from three equations ( r ) · · · + x i + · · · = · · · + x j + . . . (25) ( r m ) ( ∗ ) = · · · + x i + . . . (26) ( r m ) · · · + x j + · · · = ( ∗∗ ) . 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