ASYMPTOTICS OF RODRIGUES’ DESCENDANTS OF A...

ASYMPTOTICS OF RODRIGUES’ DESCENDANTS OF A

POLYNOMIAL

RIKARD BØGVAD, CHRISTIAN HAGG, AND BORIS SHAPIRO

Abstract. Motivated by the classical Rodrigues’ formula, we study below the

root asymptotics of the polynomial sequence

R[αn],n,P (z) =d[αn]Pn(z)

dz[αn], n = 0, 1, . . .

where P (z) is a given polynomial, α < degP is a fixed positive number, and

[αn] stands for the integer part of αn.

Our answer is expressed in terms of an explicit harmonic function deter-mined by a certain plane rational curve. This curve is obtained by means of

the saddle point method. In particular, this curve is the zero locus of the

bivariate algebraic equation satisfied by the Cauchy transform of the asymp-totic root-counting measure for the latter sequence and it is associated with an

abelian differential with imaginary periods. Additionally, we derive the linear

ordinary differential equation satisfied by R[αn],n,P (z).

Contents

1. Introduction 21.1. Main Problem 21.2. Main results 31.3. Techniques and examples 42. Various preliminaries 62.1. Basics of logarithmic potential theory 62.2. Differentials with imaginary periods 72.3. Tropical trace 92.4. Branched pushforward and piecewise-analytic forms 102.5. Affine Boutroux curves 113. The curves given by (1.3) and (1.5) are affine Boutroux curves 154. Proof of Theorem 1 205. Differential equations satisfied by Rodriques’ descendants 286. Example of a quadratic polynomial P (z) 307. Final Remarks and open problems 31References 33

Date: December 19, 2019.

2010 Mathematics Subject Classification. Primary 31A35, Secondary 12D10, 26C10.Key words and phrases. Rodrigues’ formula, successive differentiation, root-counting measures,

affine Boutroux curves.

1

2 R. BØGVAD, CH. HAGG, AND B. SHAPIRO

Figure 1. (Benjamin) Olinde Rodrigues.

1. Introduction

In 1816 (Benjamin) Olinde Rodrigues1 discovered his famous formula

Pn(z) =1

2nn!

dn

dzn((z2 − 1)n

)(1.1)

for the Legendre polynomials which undoubtedly became a standard tool in thetoolbox of the theory of classical orthogonal polynomials and special functions, seee.g. [AbSt].

Later this formula was also rediscovered by Sir J. Ivory and C. G. Jacobi, see[As]. Among other properties, the n-th Legendre polynomial Pn(z) satisfies thelinear ordinary differential equation

(1− z2)y′′ − 2zy′ + n(n+ 1)y = 0. (1.2)

1.1. Main Problem. Imitating Rodrigues’ approach, one can, for any given poly-nomial P of degree d ≥ 1, consider a double-indexed family of polynomials givenby the Rodrigues-like expression

Rm,n,P (z) :=dm

dzm(Pn(z)) , n = 0, 1, . . . and m = 0, 1, . . . , nd,

see an illustration in Fig. 3. If P = (z2 − 1) and m = n, we get the aboveclassical case of the Legendre polynomials up to a scalar factor. Analogously, fora meromorphic function f given in some open domain Ω ⊆ C, one can define its(m,n)-th Rodrigues’ descendant in Ω as

Rm,n,f (z) :=dmfn(z)

dzm.

1Born in a Jewish family of sephardic origin in Bordeaux on October 6, 1795, O. Rodrigues,thanks to Napoleon’s measures ensuring equality of rights for different religious minorities, wasable to attend Lycee Imperial which he joined in 1808 at the age of 14. Besides his mathematical

interests, he had another passion: banking and its usage for social purposes. He has been a closefriend and supporter of Saint-Simon and a very peculiar philanthropic figure with strong socialistundertones, see more details in [Al] and [AlOr].

ASYMPTOTICS OF RODRIGUES’ DESCENDANTS OF A POLYNOMIAL 3

In the present paper, we study the asymptotic root behavior for the sequencesof Rodrigues’ descendants when f is a polynomial postponing the case when f is arational function for a future publication.

1.2. Main results. In what follows, we will always assume that a polynomialP (z) under consideration satisfies the condition d := degP ≥ 2. The remainingcase d = 1 is trivial.

For any polynomial P and its Rodrigues’ descendant Rm,n,P (z), denote byµm,n,P the root-counting measure of Rm,n,P (z) and by

Cm,n,P (z) :=R′m,n,P (z)

(dn−m) · Rm,n,P (z)

its Cauchy transform. Notice that dn − m = degRm,n,P . (For the used basicdefinitions from potential theory consult § 2.2 and [Ra].)

Theorem 1. For any polynomial P and a given positive number α < degP , thereexists a weak limit

µα,P = limn→∞

µ[αn],n,P .

Moreover, its Cauchy transform Cα,P defined as the pointwise limit

C := Cα,P (z) := limn→∞

C[αn],n,P (z)

exists almost everywhere (a.e.) in C and satisfies the algebraic equation:

d∑k=0

αk−1 (α− k)(d− α)d−k

k!P (k)Cd−k = 0. (1.3)

0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 2 4 6 8 10 12Re

2

3

4

5

6

7

8

Im

0 2 4 6 8 10 12Re

2

3

4

5

6

7

8

Im

0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 2 4 6 8 10 12Re

2

3

4

5

6

7

8

Im

0 2 4 6 8 10 12Re

2

3

4

5

6

7

8

Im

Figure 2. The zeros of Rm,60,P (z) shown by the small red dots.(The larger dots are the zeros of P , the triangle is the center ofmass of the zero locus of P , and the squares are branch points of(1.3) and (1.5) in the z-plane when α = m/60.) Both in the leftand in the right subfigures, m = 3 (top left), m = 18 (top right),m = 60 (bottom left), and m = 60(degP − 1) (bottom right).

Corollary 1. The scaled Cauchy transform W defined by

W :=Wα,P :=d− ααCα,P (1.4)

satisfies a simpler algebraic equation:


d∑k=0

α− kk!

P (k)Wd−k = 0. (1.5)

Corollary 2. Equations (1.3) and (1.5) define rational affine curves which areirreducible if and only if all roots of P are simple. If P has roots of multiplicityat least 2, then (1.5) admits a finite number of “trivial” factors of the form W =(b− z)−1, where b is such a root. The remaining factor of (1.5) is irreducible andcan be written as

αW =P ′(z +W−1

)P (z +W−1)

=d logP (z +W−1)

dz. (1.6)

Remark 1. Equation (1.6) implies the following relation satisfied by the originalCauchy transform

(d− α)C =d logP (z + α

d−αC−1)

dz. (1.7)

Remark 2. Observe that, for any 0 < α < d, the support Sα,P of µα,P is containedin the convex hull of the zero locus of the polynomial P .

Corollary 3. For any polynomial P and 0 < α < d, the support Sα;P of µα;P

consists of finitely many compact semi-analytic curves and points. The measureµα;P has point masses if and only if α < 1.

Fig. 2 and 3 show several illustrations of µα;P .

1.3. Techniques and examples. Most of the above mentioned results are ob-tained as an application of a general framework which we develop in § 2.2 andwhich can be useful in various asymptotic questions involving polynomial sequencesoriginating from families of linear ordinary differential equations.

Namely, given C2 with coordinates (z, w), we define and study a special class ofplane curves which we call affine Boutroux curves; such a curve is characterized bythe fact that the standard 1-form w dz has only imaginary periods on the normal-ization of its compactification in CP 1

z ×CP 1w. Special type of Boutroux curves was

earlier introduced in [BM] where also the term “Boutroux curves” was coined. Thisnotion was further elaborated in [Be] and later used by a number of authors. Forevery affine Boutroux curve, we define a special harmonic function on this curveas well as its push-forward to CP 1

z . This push-forward is a piecewise harmonicfunction whose Laplacian (considered as a 2-current) is a signed measure on CP 1

z

supported on a finite union of semi-analytic curves and isolated points. The mostessential property of this measure is that its Cauchy transform satisfies a.e. in CP 1

z

the same algebraic equation which defines the initial affine Boutroux curve!

Let us now illustrate our results in concrete cases.

(i) For P = z2 + az + b, equation (1.3) reduces to

(2− α)(z2 + az + b)C2 + (α− 1)(2z + a)C − α = 0. (1.8)

If α = 1, and P is a monic polynomial, then


0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 2 4 6 8 10 12Re

0

2

4

6

8

Im

-10 -5 0 5Re

-6

-4

-2

0

2

4

6

Im

-10 0 10 20 30 40Re

-20

-10

0

10

Im

Figure 3. The union of all zeros of Rm,30,P (z) for m = 0, 1,. . . , 30 degP − 1 shown by small red dots. The large dots are thezeros of P , the large squares are the critical points of P , the triangleis the center of mass of the zero locus of P , and the small squaresare the branch points of (1.3) and (1.5) in the z-plane.

(ii) For d = 2, we get

PC2 − 1 = 0.

(iii) For d = 3, we get

PC3 − P ′′

2! 22C − 2

23= 0.

(iv) For d = 4, we get

PC4 − P ′′

2! 32C2 − 2P ′′′

3! 33C − 3P (iv)

4! 34= 0.

(v) For d = 5, we get

PC5 − P ′′

2! 42C3 − 2P ′′′

3! 43C2 − 3P (iv)

4! 44C − 4P (v)

5! 45= 0.

The structure of the paper is as follows. After recalling some basic notions in§ 2.1, we introduce in §§ 2.2 – 2.5.2 affine Boutroux curves and related harmonicfunctions and measures. In § 3, we prove that the algebraic curves given by (1.3)and (1.6) are affine Boutroux curves. In § 4, we settle Theorem 1 by using aspecial version of the saddle point method. In § 5, we derive linear differentialequations satisfied by Rodrigues’ descendants. In § 6, we discuss in details the case


of quadratic polynomial P . Finally, in § 7, we suggest a number of open problemsrelated to this topic.

Acknowledgements. The authors are sincerely grateful to the Mittag-Leffler insti-tute for the hospitality in June 2018 during the workshop “Hausdorff geometry ofpolynomials and polynomial sequences” where some of these results were presented.

2. Various preliminaries

2.1. Basics of logarithmic potential theory. For the convenience of our read-ers, let us briefly recall some notions and facts used throughout the text. Let µ bea finite compactly supported complex measure in the complex plane C. Define thelogarithmic potential of µ as

Lµ(z) :=

∫C

ln |z − ξ| dµ(ξ)

and the Cauchy transform of µ as

Cµ(z) :=

∫C

dµ(ξ)

z − ξ.

Standard facts about the logarithmic potential and the Cauchy transform in-clude:

• Cµ and Lµ are locally integrable; in particular they define distributions on

C and therefore can be acted upon by ∂∂z and ∂

∂z .

• Cµ is analytic in the complement in CP 1 ' C ∪ ∞ to the support of µ.For example, if µ is supported on the unit circle (which is the most classicalcase), then Cµ is analytic both inside the open unit disc and outside theclosed unit disc.• the relations between µ, Cµ and Lµ are as follows:

Cµ = 2∂Lµ∂z

and µ =1

π

∂Cµ∂z

=2

π

∂2Lµ∂z∂z

=1

2π

(∂2Lµ∂x2

+∂2Lµ∂y2

).

They should be understood as equalities of distributions.• the Laurent series of Cµ in a neighborhood of ∞ is given by

Cµ(z) =m0(µ)

z+m1(µ)

z2+m2(µ)

z3+ . . . ,

where

mk(µ) =

∫Czk dµ(z), k = 0, 1, . . .

are the harmonic moments of measure µ.

Given a polynomial p, we associate to p its standard root-counting measure

µp =1

deg p

∑i

miδ(z − zi),

where the sum is taken over all distinct roots zi of p and mi is the multiplicity ofzi.

One can easily check that the Cauchy transform of µp is given by

Cµp =1

deg p· p′

p.


For more relevant information on the Cauchy transform we will probably recom-mend a short and well-written treatise [Ga].

The above notions of a complex measure µ compactly supported in C, its log-arithmic potential Lµ, and its Cauchy transform Cµ have natural extensions tosimilar objects µ, Lµ, Cµ defined on CP 1 ⊃ C and such that the main relationsbetween these objects are preserved. They are constructed as follows.

(i) For a finite complex measure µ compactly supported in C, we introduce thesigned measure µ of total mass 0 defined on CP 1 by adding to µ the point measure−m · δ(∞) placed at ∞, where m =

∫C dµ. (It is natural to think of µ as an exact

2-current on CP 1.)(ii) The logarithmic potential Lµ is defined as a function on C ⊂ CP 1 with a

logarithmic singularity at ∞. In terms of a local coordinate w = 1/z at ∞ thelogarithmic potential is L1

loc near ∞, and this implies that we may talk about itsderivatives. In the following when we think of Lµ as an object on CP 1 we denoteit by Lµ.

Recall that on any complex manifold the exterior differential d (acting on cur-rents) is standardly decomposed as d = d′ + d′′, where d′ is the holomorphic partand d′′ is the anti-holomorphic part. For a function f on a Riemann surface witha local coordinate z, we get

d′f =∂f

∂zdz and d′′f =

∂f

∂zdz.

The above quantities µ and Lµ satisfy the relation

µ dx ∧ dy =i

πd′d′′Lµ.

More explicitly, we have that

µ dx ∧ dy =1

2π

(∂2Lµ∂x2

+∂2Lµ∂y2

)dx ∧ dy =

2

π

∂2Lµ∂z∂z

dx ∧ dy =i

π

∂2Lµ∂z∂z

dz ∧ dz,

where∂2Lµ∂z∂z is understood as a distribution on CP 1.

(iii) Finally, the Cauchy transform Cµ should be understood as an 1-current givenby the relation

Cµ = 2 d′Lµ = 2∂Lµ∂z

dz.

Then

µ dx ∧ dy =i

πd′ d′′Lµ = − i

2πd′′Cµ =

i

2π

∂Cµ∂z

dz ∧ dz.

2.2. Differentials with imaginary periods. To settle Theorem 1, we need tointroduce a special class of plane algebraic curves and show how they give riseto measures on (open subsets of) CP 1. This is a special case of a more generalconstruction which can be carried out for Riemann surfaces endowed with an abeliandifferential (i.e., a meromorphic 1-form) and a meromorphic function.

Usually multi-valued harmonic and subharmonic functions on Riemann surfacesare constructed by integrating meromorphic 1-forms. For some special types of dif-ferentials, one can get uni-valued functions instead of multi-valued. This is exactlythe situation which we want to capture. By a period of a meromorphic 1-form on


a Riemann surface Y we mean the integral of this form along a piecewise smoothclosed curve in Y .

Definition 1. A meromorphic 1-form ω defined on a compact orientable Riemannsurface Y is said to have purely imaginary periods if all its periods are purelyimaginary complex numbers.

Remark 3. Observe that the periods of ω can be roughly subdivided into twodifferent types: a) periods related to the poles of ω, i.e. integrals of ω over smallloops surrounding the poles and b) periods related to the non-trivial 1-dimensionalhomology classes of Y , i.e. integrals of ω over global cycles in H1(Y,R). (Observehowever that these two types of periods are, in general, dependent.)

Note that the first type of periods are purely imaginary if and only if all residuesof ω are real and that the second type of periods do not occur if Y has genus 0.

Remark 4. Definition 1 makes sense even if Y is not a compact Riemann surface.For our purposes, it suffices that Y is an open subset of a compact Riemann surface

Y such that Y \ Y consists of a finite number of points. This will always be thecase for e.g. smooth quasi-affine plane algebraic curves. Note that a meromorphic

1-differential ω on Y has purely imaginary periods if and only if the restriction ofω to Y has purely imaginary periods.

For a meromorphic 1-form ω with purely imaginary periods given on a compactRiemann surface Y , denote by Pol−ω ⊂ Y (resp. Pol+ω ⊂ Y ) the set of all poles ofω with negative (resp. positive) residues.

Meromorphic 1-forms (i.e. abelian differentials) with purely real periods havebeen earlier considered in e.g. [GrKr] where they were used to study the modulispaces of Riemann surfaces with marked points. (Here we consider purely imaginaryperiods, but the translation is trivial.) One of the main results of [GrKr] is asfollows, see Proposition 3.4 in loc. cit.

Proposition A. For any (Y, p1, . . . , pn) ∈ Mg,n, any set of positive integersh1, . . . , hn, any choice of hi-jets of local coordinates zi in the neighborhood of markedpoints pi, and any singular parts (i.e., for i = 1 . . . n, the choice of Taylor coeffi-

cients c1i , . . . , chii , with all imaginary residues c1i ∈ iR and the sum of the residues∑

c1i vanishing), there exists a unique differential Ψ on Y with purely real periodsand prescribed singular parts. In other words, in a neighborhood Ui of each pi thedifferential Ψ satisfies the condition

Ψ|Ui =

hi∑j=1

cjidz

zj+O(1).

Proposition A implies that on an arbitrary compact Riemann surface Y thereexists a large class of meromorphic 1-forms with purely imaginary periods. Furtherwe can associate to each meromorphic differential with imaginary periods on Y areal-valued function Y \Polω → R. Namely, fix a point p0 ∈ Y \Polω and considerthe multi-valued primitive function

Ψ(p) :=

∫ p

p0

ω.

Ψ(p) is a well-defined uni-valued function on the universal covering of Y \ Polω.The next statement is trivial.


Lemma 2. In the above notation, ω has purely imaginary periods if and only if themulti-valued primitive function Ψ(p) has a uni-valued real part Re Ψ(p). In otherwords,

Hω(p) := Re Ψ(p)

is uni-valued already on Y \ Polω.

Note that Hω(p) is continuous and harmonic in a neighborhood of any point inY \ Polω. The local behavior of Hω around a pole p is determined by the sign ofthe residue of ω at this point. Namely, let ω be a meromorphic 1-form with purelyimaginary periods and only simple poles. For p ∈ Polω, let z be a local coordinateat p, and denote by c the residue of ω at p.

Lemma 3. In the above notation, Hω is a subharmonic L1loc-function on Y \Pol−ω .

Locally, for the restriction of Hω to a suitable neighborhood of p, the following holds:

(1) Hω(z) = c log |z| + Hω(z), where Hω(z) is harmonic in a neighborhood of

p. Consequently, ∂2Hω∂z∂z = ( cπ2 )δp, where δp is the Dirac measure at p, and

the derivatives are taken in the sense of distributions.(2) If p ∈ Pol+ω , then there is a neighborhood of p in which Hω is a well-defined

subharmonic function. As such, Hω(p) = −∞.(3) If p ∈ Pol−ω , then limz→pHω(z) =∞.

(4) ∂Hω(z)∂z = 1

2ω.

Proof. (1) is a consequence of the fact that ω has a simple pole at p and hencelocally it can be written as ω = c

z dz + ω, where ω is holomorphic at p. Then (2)and (3) follow, while (4) follows from the standard relation

∂Ψ

∂z= ω = 2

∂(Re Ψ)

∂z.

2.3. Tropical trace. Given a branched covering ν : Y → Y ′ of Riemann surfacesand a function f : Y → R, we will define the induced function on Y ′ by takingthe maximum of the values of f on each fibre (instead of the summation or theintegration over the fiber as it is done in the case of the usual trace). It seems thatthis construction which frequently occurs in the study of the root asymptotics forsequences of polynomials has not been given any special name yet.

Definition 2. Given a branched covering ν : Y → Y ′ and a real-valued functionf : Y → R, we define its tropical trace ν∗f : Y ′ → R as

ν∗f(z) = maxqi∈ν−1(z)

f(qi).

The same definition extends to real-valued functions f defined on Y \S, where S isa discrete set such that for any s ∈ S, limz→s f(z) exists as a real number or ±∞.(In other words, we allow f to attain values ±∞.)

Example 1. Let Hi, i ∈ 1, 2...., n := [n] be an n-tuple of real harmonic functionson Y ′. They define a harmonic function H on Y := Y ′ × [n] by

H(z, i) = Hi(z).


For the canonical projection ν : Y → Y ′ given by ν(z, i) = z, we get

ν∗H(z) = maxi∈[n]

Hi(z).

Furthermore, the Laplacian of ν∗H is supported on the union of the real semi-analytic curves Hi(z) = Hj(z), i 6= j. In fact, this Laplacian considered as ameasure is explicitly given by the Plemelj-Sokhotski formula, see e.g. [BB].

Definition 2 works for an arbitrary finite map between complex manifolds. Theelementary fact that the maximum of a finite number of subharmonic functions issubharmonic, implies certain restrictions on the support of the Laplacian of ν∗fwhich makes Definition 2 useful. In particular, the tropicaltrace of a subharmonicfunction is subharmonic. We describe this in more detail in Theorem 4 below.

Let Y be a (not necessarily compact) Riemann surface and let ν : Y → Y ′ be abranched covering. Let f : Y → R be a real-valued function harmonic except for afinite number of points P where it has logarithmic singularities. In other words, in

a neighborhood of a singular point p ∈ P , f(z) = c log |z|+ f(z), where z is a local

coordinate and f(z) is harmonic. Let P− (resp. P+) be the set of those p ∈ P atwhich the residue c is negative (resp. c > 0). Then f is subharmonic in Y \ P−.Note that P supports all the point masses of the Laplacian of f considered as ameasure.

Theorem 4. Under the above conditions, the tropical trace ν∗f is harmonic in theopen set Y ′ \ν(P ), subharmonic in U = Y ′ \ν(P−), and it has at most logarithmicsingularities. The Laplacian of ν∗f in U is supported on a finite union of real semi-analytic curves and points; the latter set is contained in the set of the images of allpoles of f under ν.

Proof. Note first that the maximum h(z) = max fi(z) of a finite number of har-monic functions fi, i = 1, ..., r, defined on an open set V ⊂ Y ′ is subharmonic;its Laplacian is supported on (some parts of) the level curves fi = fj , i 6= j.Furthermore these level curves are real semi-analytic.

In such a situation h(z) is subharmonic. Only in the case when fi(p) = −∞ forall i = 1, 2, ..., r, one gets h(p) = −∞. In this situation, in addition to a possiblemeasure supported on a real semi-analytic curve, the Laplacian of h will have apoint mass (mini ci)δp at p.

If some ci < 0, then h will not be subharmonic at p, but still it has a logarithmicsingularity there and its Laplacian contains a negative point mass at p.

Let BR ⊂ Y denote the set of all branch points of ν and set D := ν(BR) ⊂ Y . IfV ⊂ Y ′ \D is a simply connected open set, there exists a disjoint union ν−1(V ) =∪V ′i , such that the restriction ν : V ′i → V is a locally biholomorphic map which wedenote by νi. Then ν∗f(z) := maxi f(ν−1

i (z)), z ∈ V is a subharmonic function,and by the previous example, its Laplacian is supported on a union of some realsemi-analytic curves and (possibly) at some points in ν(P+). A similar argumentworks in the case p ∈ D; in suitable local coordinates on Y and Y ′ resp., ν canbe written as w 7→ z = ν(w) = wr. The rest of Lemma follows since it is alwayspossible to cover Y ′ \D by a finite number of open simply-connected sets such thatthe preceding argument holds.

2.4. Branched pushforward and piecewise-analytic forms.


Definition 3. Given a branched covering ν : Y → Y ′, where Y and Y ′ are compactRiemann surfaces, by a uni-valued branch of ν we mean an open subset U ⊂ Y suchthat ν maps U diffeomorphically onto its image ν(U) and additionally, ν(U) = Y ′\C,where C is a finite union of smooth compact curves and points in Y ′.

An easy way to get several uni-valued branches simultaneously is to fix a cutC ⊂ Y ′ such that(i) C contains all the branch points of ν;(ii) Y ′ \ C consists of contractible connected components.

Then the surface Y \ ν−1(C) splits into d disjoint uni-valued sheets which areuniquely defined in case when Y ′ \ C is connected.

Definition 4. Given a meromorphic 1-form ω on a compact Riemann surface Yand a branched covering ν : Y → Y ′ of some degree d, where Y ′ is a compactRiemann surface, we define the branched push-forward ν∗ω as a d-valued 1-formon Y ′ obtained by assigning to a tangent vector v at any point p ∈ Y ′ \Dν one of dpossible values ω(ν−1

j v), j = 1, . . . , d. Here ν−1(v)j is one of d possible pull-backsof v to the tangent bundle of Y and Dν is the set of all discriminantal points ofν. (Observe that ν is a local diffeomorphism near any point of Y which is not acritical point.)

Using fancier language, the above definition means that we are considering a (set-theoretic) section θ : Y ′ → Y of the covering ν : Y → Y ′, which chooses at eachpoint in Y ′ (with some finite number of exceptions) one of the d possible points inthe fiber. This operation induces a branched push-forward ν∗ω as a (set-theoretical)section of the bundle of meromorphic 1-forms on Y ′. We use set-theoretical sections,since we are not requiring any differentiability of θ.

We want θ to satisfy the condition similar to the one which results from providinga cut of Y ′ in order to obtain a d-fold covering by disjoint sheets, as described above.In other words, we want to remove from Y ′ a cut which is a set of Lebesgue measure0, and decompose the remaining surface into open domains on each of which θ isbiholomorphic. More precisely, assume that Y ′ \Dν = ∪ni=1Ui ∪ E, where Uis aredisjoint open sets and E has Lebesgue measure 0 such that θ is biholomorphic oneach Ui and is a section of ν. In this case we will say that the associated ν∗ω ispiecewise-analytic.

The Cauchy transform of the asymptotic root-counting measure which we willconstruct later will be piecewise-analytic in the above sense. Recall that we thinkof the Cauchy transform as a 1-current, i.e. as a generalized 1-form. The piece-wise analytic character of our construction stems from the fact that the Cauchytransform will be associated to a section of a finite cover.

2.5. Affine Boutroux curves. Consider an irreducible affine algebraic curve Γ ⊂Cz × Cw, where the product Cz × Cw ' C2 is equipped with a coordinate system(z, w).

Denote by Γ ⊂ CP 1z × CP 1

w the closure of Γ and denote by Γ the normalisation

of Γ. Denote by πz : Cz × Cw → Cz or, similarly, by πz : CP 1z × CP 1

w → CP 1z

the standard projection onto the first coordinate and by πw the projection onto thesecond coordinate.


Denote by n : Γ → Γ the standard normalisation map. Recall that the smooth

compact Riemann surface Γ is birationally equivalent to Γ. Now consider on Cz×Cw(resp. on CP 1

z × CP 1w) the standard meromorphic 1-form

Ω := w dz.

One can easily show that the zero divisor of Ω on the latter space is a copy of CP 1

given by w = 0 which we denote by H0w; (the closure of) its pole divisor is the union

of two intersecting copies H∞z ∪H∞w of CP 1 given by w =∞ and z =∞.Given a curve Γ ⊂ Cz × Cw as above, consider the meromorphic 1-form

ΩΓ := Ω|Γ resp. ΩΓ := Ω|Γobtained by the restriction of Ω to Γ (resp. to Γ). Denote by Ω the pullback of ΩΓ

to Γ under the normalisation map n. This form will be the key ingredient below.

The zero divisor of ΩΓ consists of the intersection points H0w ∩ Γ and all singu-

larities of Γ ⊂ CP 1z × CP 1

w.The pole divisor of ΩΓ consists of non-singular points of the intersection

(H∞z ∪H∞w ) ∩ Γ.

Given an irreducible affine curve Γ ⊂ Cz × Cw as above and the corresponding

meromorphic 1-form Ω on Γ, consider the multi-valued primitive function

Ψ(p) =

∫ p

p0

Ω.

Ψ(p) is a well-defined uni-valued function on the universal covering of Γ\Pol, where

Pol ⊂ Γ is the set of all poles of Ω and p0 ∈ Γ \ Pol is some fixed base point. Thenext statement is trivial, cf. 2.2.

Lemma 5. In the above notation, Ω has purely imaginary periods if and only if themulti-valued primitive function Ψ(p) has a uni-valued real part Re Ψ(p). In other

words, Re Ψ(p) is uni-valued already on Γ \ Pol.

Definition 5. A plane affine irreducible curve Γ ⊂ Cz × Cw is called an affine

Boutroux curve (aBc, for short) if its associated meromorphic 1-form Ω has purely

imaginary periods on Γ.

We can reformulate the latter definition as follows. Let Γsm ⊆ Γ be the smoothpart of Γ. Then Γ is an aBc if and only if the restriction of w dz to Γsm has purelyimaginary periods on Γsm. In fact, this is equivalent to the requirement that w dz

has purely imaginary periods on any smooth Γ1 ⊆ Γ such that Γ \ Γ1 is finite.

2.5.1. How to construct affine Boutroux curves. The following result describes away to obtain affine Boutroux curves. In Section 3 we will prove that the curve (1.3)is an aBc curve and show that this curve is an instance of our present construction.

We start with a real-valued harmonic function H(z, w) on Cz×Cw\S, where S isa finite set, such that the holomorphic differential d′H = P dz+Q dw has rationalcoefficients. Write P = P1/R1 and Q = Q1/R2 with relatively prime polynomialnumerators and denominators. Let R be the least common multiple of R1 and R2

and assume that

(*) Q1 /∈ C[z] is irreducible and relatively prime to R.


Denote by U ⊂ Cz × Cw the complement to the zero locus of R. Let D ⊂ U ⊂Cz × Cw be a curve given by Q = 0, and let πz : D → Cz be the (restrictionof the) projection sending (z, w) → z. The affine ring of functions on U coincideswith the ring C[z, w]R of rational functions whose denominators are powers of R.The affine ring of functions on D is the domain

C[z, w]R/(Q) = (C[z, w]/(Q1))R.

By (*), Q1 is a non-constant function no factor of which is a polynomial in thesingle variable z and is also relatively prime with respect to R. Therefore the mapsof commutative rings

C[z] ⊂ C[P, z]R(Q)

⊂ C[z, w]R(Q)

are inclusions and induce a factorization πz = π1 π2 given by

D → C := specC[P, z]R

(Q)→ C = spec C[z]

by maps with finite fibers. Consider the map ν : C → C2 induced by the mapC[u, z]→ C[P, z] sending u→ P .

Proposition 6. In the above notation, the curve C is an aBc.

Proof. Since Q1 is an irreducible polynomial, C is an irreducible affine variety.Except for a finite number of points S ⊂ C, both z and u are local coordinateson C and ν is a covering map. By the remark following the definition of affineBoutroux curves, it suffices to check that the periods of w dz on the complementC \S are purely imaginary. Let H be the restriction of H(w, z) to C, and note thatd′H = P dz = 1

2u dz. Proposition 6 follows from the fact that the real part of aprimitive function of w dz (which is well-defined on the universal cover) is actuallygiven by the uni-valued function H defined on C.

Theorem 7 below will explain our interest in affine Boutroux curves.

2.5.2. Affine Boutroux curves, induced signed measures on CP 1, and their Cauchytransforms. In this subsection we will show that under some additional conditions,given an affine Boutroux curve Γ ⊂ Cz × Cw, we can construct a signed measureon CP 1

z whose Cauchy transform satisfies the algebraic equation defining Γ.

Indeed, given a plane curve Γ ⊂ Cz ×Cw as above, we have a natural meromor-

phic function τ : Γ → CP 1z induced by the composition of the normalisation map

n : Γ → Γ with the projection of πz : Γ ⊂ CP 1z × CP 1

w along the w-axis onto thez-axis. The next result is our main technical tool.

Theorem 7. In the above notation, given an aBc Γ ⊂ Cz × Cw, such that therestriction of w dz to Γ has at most simple poles, there exists a signed measure µΓ

of total mass 0 supported in CP 1z with the following properties:

(i) its support Sµ := supp(µΓ) ⊂ CP 1z consists of finitely many compact real semi-

analytic curves and isolated points;

(ii) its Cauchy transform CµΓ(considered as a 1-current, see the end of § 2.1)

coincides with a uni-valued piecewise analytic branch of τ∗Ω in CP 1z \ Sµ. In other

words, if we represent CµΓ= C(z) dz in the affine chart Cz ⊂ CP 1

z , where C(z) is adistribution, then C(z) satisfies in Cz \ Sµ the algebraic equation defining Γ;


(iii) the support S−µ of the negative part of µΓ coincides with τ (Pol−) ⊂ CP 1z .

Remark 5. Observe that the converse to the statement of Theorem 7 is, in general,false, i.e. there exist curves for which conditions (i), (ii) and (iii) still hold butwhich are not affine Boutroux curves, see e.g. § 4 of [BoSh]. Thus being an aBconly provides a sufficient condition for (i) – (iii). Observe additionally that if weremove condition (iii), then there exists situations in which µΓ is not unique, see e.g.Theorem 4 of [STT]. We also want to point out a close connection of Theorem 7with some results of [BaSh] where condition (iii) is described as the existence ofclean poles.

Proof. Choose an arbitrary point p0 ∈ Γ \ Pol and, as in Lemma 3, consider thefunction

Re Ψ(p) = Re

[∫ p

p0

Ω

],

where Ω = w dz. Since Γ is an aBc, then Ω has purely imaginary periods and

Re Ψ(p) is a uni-valued harmonic function on Γ \ Pol. (One can consider Re Ψ(p)

as defined on the whole Γ if one allows it to attain values ±∞.) Let BR be the

branch locus of the projection τ : Γ→ CP 1z and τ(Pol) be the image of the set Pol

of all poles of ΩΓ. Here τ = n πz. Recall that the branching locus BR ⊂ CP 1z is

defined as the set of all critical values, i.e. all branch points of the meromorphic

function τ : Γ→ CP 1z . Obviously, BR is a finite set.

Now, for any z ∈ CP 1z \ (BR ∪ τ(Pol)), define the function Θ on CP 1

z given by

Θ(z) := maxqi∈τ−1(z)

Re Ψ(qi).

In other words, Θ(z) is the tropical trace of the projection τ of Re Ψ to CP 1z .

Observe that if z lies in CP 1z \ (BR ∪ τ(Pol)), then it is a local parameter on

every branch of Γ near each point belonging to the fiber τ−1(z), which impliesthat each function Hi(z) := Re 2Ψ(pi) is a well-defined harmonic function near z.Moreover, outside of its poles, Θ(z) is a continuous subharmonic function.

The above definition of Θ(z) also makes sense if z is a branch point or the image

of a pole; in the latter case Θ(z) might attain infinite values. Namely, if p ∈ Γ is apole with residue κi and zp = τ(p), then locally near zp the corresponding Hi(z) hasthe asymptotics κi log |z − zp|. Hence, if κi is positive, then limz→zp Hi(z) = −∞in a sufficiently small neighbourhood of zp. Analogously, if κi is negative, thenlimz→zp Hi(z) = +∞. Finally, Θ(z) = −∞ if and only if every point in τ−1(z) is a

pole of Ω with a positive residue.

Now define the 2-current µΓ on CP 1z given by

µΓ :=1

2π

(∂2Θ

∂x2+∂2Θ

∂y2

)dx dy =

i

π

∂2Θ

∂z∂zdz dz, (2.1)

where (x, y) are the real and the imaginary parts of the affine coordinate z. Wewill call the function Θ the logarithmic potential of the 2-current µΓ.

The 2-current µΓ given by (2.1) satisfies conditions (i)-(iii) of Theorem 7. Con-ditions (i) and (iii) follow from Lemma 4, which says that µΓ is actually a signedmeasure on CP 1 supported on finitely many semi-analytic curves belonging to the


level sets Re Ψ(pi) = Re Ψ(pj), i 6= j and finitely many isolated points includingτ(Pol−) and possibly some part of τ(Pol+).

For (ii), assume that V is a connected and simply connected subset of the com-plement U to supp(µΓ) ∪ BR∪ τ(Pol). Then Θ(z) = Ψ(pV (z)) for a certain choiceof a branch pV : V → Γ ⊂ Cz × Cw. Set pV (z) = (z, w(z)) where w is a branch ofthe algebraic function defined by Γ. Clearly z is a local coordinate both in V andin pV (V ) and therefore

d′(Θ(z)) =∂Θ(z)

∂zdz = w(z) dz.

Observe that since µΓ has a potential, it must necessarily be exact, i.e.∫CP 1

µΓ = 0.

By construction, the negative part of µΓ is supported on τ(Pol−).

3. The curves given by (1.3) and (1.5) are affine Boutroux curves

To prove Theorem 1 using the above Theorem 7, we need to study in detail thealgebraic curves given by (1.3) and (1.5). In what follows we will denote

(i) by Γα,P ⊂ Cz × CC the affine algebraic curve given by (1.3).

Our goal is to prove that any irreducible Γα,P is an affine Boutroux curve (aBc).In fact, we will prove this property for the curve Λα,P defined by

(ii) Λα,P ⊂ Cz × CW is the affine algebraic curve given by (1.6).

Since Γα,P is obtained from Λα,P by a real scaling of the first coordinate, theclaim that Γα,P is an aBc follows from that for Λα,P .

Remark 6. Observe that (1.3) defines the closure Γα,P in CP 1z × CP 1

C of thebidegree (d, d). By the adjunction formula, a smooth curve in CP 1

z × CP 1C of the

bidegree (d, d) has genus (d−1)2. However, the curve given by (1.3) is rational andtherefore highly singular.

The next technical theorem describes the algebraic geometric properties of thesymbol curve and its canonical differential W dz which are central for the appli-

cation of the tropical trace to our problem. In Theorem 8, Λ stands for Λα,P , Λ

for Λα,P which is the closure of Λα,P in CP 1z × CP 1

W , and Λ for Λα,P which is the

normalisation of Λα,P . Recall that W = d−αα C is a coordinate obtained by scaling

of the coordinate C.

Theorem 8. Let P be a polynomial of degree d ≥ 2 and let 0 < α < d be a positivenumber. Assume that P and P ′ have only simple zeros. Then the algebraic curveΛ ⊂ Cz × CW given by (1.6) is an aBc.

More exactly, the following properties hold:

(i) Λ is an irreducible rational curve.

(ii) The inverse image π−1z (∞) ⊂ Λ ⊂ CP 1

z × CP 1W consists only of (0,∞), that is

∞ ∈ CP 1z is a complete ramification point of the function τ : Λ→ CP 1

z .

(iii) The equation defining the slopes v of different branches of Λ at ∞ ∈ CP 1z is

given by

(v + 1)d−1(α(v + 1)− d) = 0


and, therefore, gives the essential slope v = dα − 1 and another slope v = −1 of

multiplicity d− 1.

(iv) The only singularity of Λ ⊂ CP 1z × CP 1

W is (0,∞). As a consequence, the

normalisation map n : Λ → Λ is 1 − 1 at all points except for (0,∞) ∈ Γ whose

preimage consists of d points of Λ.

(v) All d local branches of Λ at the point (∞, 0) are smooth.

Finally, the set of all poles of Ω is presented in (vi)-(vii) below.

(vi) Ω has a simple pole at each of the points pi ∈ Λ, i = 1, . . . , d whose image

n(pi) = (∞, zi) ∈ Λ ⊂ CP 1z × CP 1

W , where zi runs over the set of zeros of P . At

each such point pi the residue of Ω equals (1− α)/α.

(vii) Ω has a pole with a real residue at each of the d preimages of the singular

point (0,∞) ∈ Λ under the normalization map n : Λ→ Λ. This residue equals 1 foreach of the d− 1 preimages coming from the branches with the slope −1 at (0,∞)and the remaining residue for the preimage coming from the essential branch equals1− d

α < 0.

Proof. To prove (i), observe that the global rational change of variables W =W, z = z +W−1 transforms (1.6) into

αW =P ′(z)

P (z). (3.1)

Since α 6= 0, this equation allows us to consider W as the graph of a rationalfunction in the variable z. The latter fact implies that Λ is a rational curve, and inaddition, Λ is irreducible since it is a graph.

To prove (ii), we argue as follows. Assuming that P (z) = (z−z1) . . . (z−zd) hasall simple zeros, we obtain

αW =P ′(z +W−1)

P (z +W−1)=

d∑i=1

1

z +W−1 − zi. (3.2)

Substituting z = 1y in (3.2) and clearing the denominators, we get

αP = y

d∑j=1

Pj , (3.3)

where

P =

d∏i=1

(y +W − ziyW), Pj =P

y +W − zjyW.

To obtain the fiber over z = ∞ ∈ CP 1z , i.e. over y = 0, one should substitute

y = 0 in (3.3). One can easily check that the result of this substitution is αWd = 0,implying that the only point in the fiber τ (−1)(∞) is W = 0. The argument works,even if P does not have simple zeros.

To settle (iii), we need to calculate the slopes of the branches of Λα,P at (0,∞),for which one should substitute W = v(y)y in (3.3). The slope coincides with


v := v(0). After the substitution of W = v(y)y in (3.3), one cancels out the factorydegP in both sides and then sets y = 0. The result is as follows

α

d∏i=1

(1 + v) = d

d−1∏i=1

(1 + v)

or, equivalently,

α(1 + v)d = d(1 + v)d−1 ⇔ (1 + v)d−1(α(v + 1)− d) = 0,

which is the required statement.

To prove (iv), we need to show that there are no singularities of Λ above theaffine part of CP 1

z , i.e. for all z 6= ∞ and W ∈ CP 1. Notice first that W = 0 isimpossible for finite z. W = 0 is equivalent to W−1 =∞. Rewrite (3.2) as

G(W−1, z) = αP (z +W−1)−W−1P ′(z +W−1) = 0. (3.4)

A calculation shows that the coefficient at the highest power of W−1 is α − d,which is non-zero by the assumptions of the theorem. Hence, z finite and W = 0

is impossible. In other words, the curve Λ intersects the coordinate line CP 1z only

at z =∞, and the part of Λ in the plane Cz × CW is contained in the open set Ugiven by W 6= 0.

Secondly, observe that the rational change of coordinates (W, z) 7→ (W, z) wherez = z +W−1 is a diffeomorphism between U : W 6= 0 in Cz × CW and V : W 6= 0in Cz ×CW . The curve given by (3.1) in the coordinates (z,W) is clearly smoothwhen z is not a root of P . Additionally,W =∞ at any root of P implying that ourcurve is smooth in the whole of V . Any diffeomorphism preserves the smoothness

property, and hence Λ is smooth in U . By the first observation, it is smooth at allpoints in Cz × CW .

It remains to check the points of Λ with W = ∞, which occurs exactly atthe roots of P . This can be verified by setting W−1 = 0 in (3.2). Assume thatp ∈ CP 1

z × CP 1W given by W−1 = 0 and z = zi is a singular point of our curve

where P (zi) = 0. Then at p, the partial derivatives of the left-hand side G(W−1, z)of (3.4) with respect to the variables W−1 and z must vanish. A short calculationleads to

∂G(W−1, z)

∂W−1− ∂G(W−1, z)

∂z= −P ′(W−1, z).

Since W−1 = 0 and P (zi) = 0 at p and we have assumed that P (z) has only simpleroots, we get that P ′(zi) 6= 0 which implies that the latter difference between thepartial derivatives can not vanish at p, a contradiction.

To prove (v), we first consider the essential branch at ∞, i.e. the branch whoseslope is given by v = d

α − 1. By our assumption, 0 < α < d, which, in particular,implies that this slope differs from −1 which is the slope for all other branches. Bythe implicit function theorem, the essential branch is smooth at (0,∞).

Let us now consider the remaining cases for which

W = yv(y) = y(−1 + v1y + v2y2 + . . . ) = −y + y2u(y). (3.5)

We will first show that if P ′ has simple roots, then there are d−1 distinct solutionsfor v1. Rewriting the equation in terms of v(y) corresponds to the blow-up of thecurve (at the origin), and then rewriting it in terms of u = u(y) corresponds to stillanother blow-up.


Note that

y−1 +W−1 =u

yu− 1,

and so substituting (3.5) in the equation (3.2) results in

αy =

d∑i=1

1

u− zi(yu− 1). (3.6)

If we insert y = 0, this equation becomes

0 =

d∑i=1

1

u+ zi= −P

′(−u)

P (−u). (3.7)

NowP ′(u)

P (u)= 0 ⇐⇒ P ′(u) = 0.

The last equation has degree d − 1 in u and its solutions are exactly the zeros ofP ′(u). Thus there are d− 1 solutions u(0) = v1 to (3.7), and these are distinct bythe assumption that P ′ has only simple roots. We can additionally observe that heequation (3.6) defines a curve αy = F (u, y) in Cu × Cy with coordinates u and y.This curve will be smooth and transversal to y = 0 at a point (v, 0) if

∂F (u, y)

∂u

∣∣(vi,0)

=

d∑i=1

1

(v + zi)2=

(P ′(−v)

P (v)

)′6= 0.

On the other hand,(P ′(−v)

P (v)

)′= −P

′′(−v)P (−v)− (P ′(−v))2

P 2(−v)= −P

′′(−v)P (−v)

P 2(−v)6= 0,

if we assume that v = v1 is one of the distinct roots of P ′(−v).This argument proves that in a neighbourhood of y = 0 there are d− 1 branches

of the affine curve (3.6) in Cu × Cy with coordinates (u, y), intersecting y = 0 inthe d − 1 different smooth points (vi, 0), where each vi is a root of P ′(−v). If weconsider these branches in CW × Cy with coordinates (W, y) using the coordinatechange Θ : (u, y) 7→ (W, y) = (−y + y2u, y), then an easy calculation shows thatthey will be mapped onto all d− 1 distinct branches all of which having the slope−1. This proves that these branches are smooth at y = 0. Note that Θ is thecomposition of two blow-ups: (u, y) 7→ (−1 + yu, y) = (u, y)) (blowing up the point(−1, 0)) and (u, y) 7→ (uy, y) (blowing up the origin (0, 0)). We have deduced thedesired results from the strict transform given by (3.6).

Thus besides the smooth essential branch, we get (d − 1) additional smoothbranches at the complete ramification point with the same slope −1 and distinctcoefficients at y2.

To prove (vi), observe that for z 6= ∞, the poles of the 1-form Ω (which is the

pullback to Λ of the formW dz restricted to Λ under the normalisation map n) occur

at (the pullbacks of) the non-singular points of Λ∩H∞W , where H∞W ⊂ CP 1 ×CP 1

is given by W = ∞. Since W = ∞ corresponds to W−1 = 0 and α 6= 0, then for

z 6= ∞, we immediately observe from (3.2) that the poles of W dz restricted to Λoccur at the points (∞, zi), where zi is any root of P .


Using (3.2), let us calculate the residues of W(z) dz restricted to Λ at each(∞, zi). Dividing equation (3.2) by W(z) and introducing the local coordinateξi = z − zi, we get

α =W−1(ξi)

W−1(ξi) + ξi+∑j 6=i

W−1(ξi)

W−1(ξi) + ξi − (zj − zi).

Assuming that W−1(ξi) = κiξi + . . . and taking the limit of the right-hand sideof the latter equation when ξi → 0, we get

α =κiξi

κiξi + ξi=

κiκi + 1

which immediately implies κi = α1−α . Thus

W(ξi) =1− ααξi

+ . . . ⇒ Res|(∞,zi) W dz =1− αα

.

To settle (vii) and to study the behavior of Ω = W dz at the singular point

(0,∞), we need to change the variable z = 1y . Then Ω = −W dy

y2 . Observe that

under the assumptions of (v), each local branch of Λ at (0,∞) is smooth whichimplies that the normalisation map is a local diffeomorphism of the corresponding

small neighborhood of Λ with this branch. Then we have the following expansionof W(y) for each local smooth branch with the slope −1 and the residue of W dzrestricted to this branch:

W(y) = −y + . . . , ⇒ −W(y)dy

y2=

(1 + . . . ) dy

y⇒ Res |(∞,0)

(−W(y)

dy

y2

)= 1.

Analogously, for the essential branch whose slope equals dα − 1, we get

W(y) =

(d

α− 1

)y + · · · ⇒ Res|(∞,0)

(−W(y)

dy

y2

)= 1− d

α.

Remark 7. Under the above assumptions of (vi), (vii), the total number of poles of

Ω equals 2d = 2 degP , all having real residues. Observe that in case of simple zerosof P , the point (0,∞) reduces the genus by (d− 1)2 which means that this point isa rather complicated singularity. Under the above assumptions, the cardinality ofCR equals 2d − 2 which implies the standard identity for the Euler characteristic

of CP 1 that the number of poles minus the number of zeros of Ω equals 2. (Polesand zeros are counted with multiplicities.)

Remark 8. The sum of all residues of any meromorphic form on any compactRiemann surface must vanish. Our count gives the following sum

Σ =1− αα· d+ d− d

α= 0.

Remark 9. Observe that the equation (1.3) defining the curve Γα,P belongs to theclass of the so-called balanced algebraic functions defined in [BoSh], § 3. For bal-anced algebraic functions it has been conjectured in § 3 of [BoSh] that there alwaysexists a probability measure with required properties. However not all balancedalgebraic functions correspond to affine Boutroux curves.


4. Proof of Theorem 1

We will use the classical saddle point method as presented in [Os], see also [Bi],§ 7.3.11 and [Br]. Let P be a monic polynomial, as in Theorem 1, and let µn bethe root-counting measure of the Rodrigues’ descendant

qn(z) := R[αn]−1,n,R(z) = (Pn)([αn]−1)(z),

where α ∈ (0, d). (Note that the order of derivation here is by one less than in theIntroduction, but this will not effect the final result.)

The sketch of the proof of Theorem 1 is as follows. For any z0 ∈ C, Cauchy’sformula for high order derivatives gives

qn(z0) =([αn]− 1)!

2πi

∫c

Pn(z) dz

(z − z0)[αn], (4.1)

where c is any simple closed curve in the z-plane encircling z0 once counterclockwise.(Notice that P has no poles.)

The saddle point method allows us to analyze the asymptotics of (4.1) whenn→∞. The degree of the polynomial qn(z) is given by dn := dn− [αn]+1. We areable to find the limit of the sequence Lµn(z) of the logarithmic potentials, where

Lµn(z) := log |qn(z)/an|1/dn and an is the leading coefficient of qn(z). It turns outthat the saddle points of the integrand of (4.1) are given by the algebraic equation(1.3). This fact enables us to identify the limit Ψ(z) := limn→∞ Lµn(z) with thetropical trace of a natural harmonic function on the associated plane curve. Finally,applying the Laplace operator, we prove that µ := limn→∞ µn exists and obtainthe algebraic equation for its Cauchy transform. Let us now provide the relevantdetails dividing them in a number of subsections.

4.0.1. Root asymptotics via the saddle point method. Given α > 0, set

n =[αn]

α+ sn, (4.2)

where 0 ≤ sn < 1/α and set m := [αn]. Consider the integral

IP =

∫γ

Pn(z) dz

(z − z0)[αn]

over a (sufficiently short) curve segment γ. Take a neighboorhood U of γ and asuitable choice of a branch of the logarithm, so that (P 1/α)mP sn = Pn in U , wherewe have used the logarithm to define powers. Then,

IP (m, sn, γ) :=

∫γ

(P 1/α(z)

z − z0

)mP sn(z) dz =

∫γ

ek(z)mP sn(z) dz, (4.3)

where

k(z) =1

αlogP (z)− log(z − z0). (4.4)

Clearly k(z) is holomorphic if γ avoids the zeros of P and z0. A saddle point is azero of k′(z).

The version of the saddle point method which we use here is as follows, seeTheorem 1.2 and Corollary 1.4 of [Os]. Assume that

(i) k(z) is any function holomorphic in a neighbourhood U of a simple curve γ;(ii) u0 ∈ γ is a saddle point of k(z) lying in the interior of γ;(iii) ∀z ∈ γ, such that z 6= u0, Re k(z) < Re k(u0).


Let ` ≥ 2 be the order of the saddle point u0, i.e.

k(z) = k(u0)− k0(z − u0)`(1− φ(z)), (4.5)

where k0 6= 0, and φ(z) is a holomorphic function in a neighbourhood of u0 suchthat φ(u0) = 0. Furthermore, assume that logP (z) is defined in U .

Lemma A. For m ∈ N and 0 ≤ s ≤ A <∞, consider

I(m, s, γ) :=

∫γ

ek(z)mP s(z) dz.

Then,

I(m, s, γ) = ek(u0)m

(Γ(`−1)

α0(ε1 − ε2)

m1`

+O

(K(P )

m2`

)), (4.6)

where ε1 and ε2 are two distinct `-th roots of unity depending only on γ, and K(P )is an upper bound for |P s(z)| in U . The constant α0 is given by

α0 =1

`· P−1/`

0 (P s(u0))

and the implicit constant in the remainder term O(...) is independent of P , s andm.

Under the hypothesis of Lemma A, we obtain the following immediate result.

Corollary A.limm→∞

|I(m, s, γ)|1/m = eRe k(u0), (4.7)

uniformly in 0 ≤ s ≤ A.

Proof. The uniformity in s follows from the fact that, by definition, K(P ) is uni-formly bounded in s. The rest is a standard limit argument.

Hence, for any sequence 0 ≤ sm ≤ 1/α,

|IR(m, sm, γ)|1/m → eRe k(u0), (4.8)

if the Cauchy integral (4.3) and its saddle point u0 satisfy the assumptions (i)–(iii)above.

4.0.2. Deformation of the contour. The saddle points of k(u) in (4.4) are those ufor which

k′(u) =P ′(u)

αP (u)− 1

u− z0= 0. (4.9)

Note that (4.9) implies that saddle points associated to z0 are independent of thechoice of a branch of the logarithm. For each z0, there are at most d = degP suchsaddle points, since the polynomial P ′(u)(u − z0) − αP (u) has degree d for anyα < d.

For all pairs (u, z) ∈ C2 except for those for which either P (u) = 0 or u = z,define

G(u, z) :=1

α(log |P (u))| − α log |u− z|) . (4.10)

Observe thatG(u, z0) = Re k(u) (4.11)

and G(u, z0) is a harmonic function of u.The integration contour for the integral (4.1) can be freely deformed as long as

it does not pass through z0.


4.0.3. Properties of the saddle point curve. The saddle point curve

D ⊂ CP 1u × CP 1

z ,

is determined by (4.9) in the affine (u, z)-plane and extended by taking the closureto CP 1

u×CP 1z . Consider the birational transformation Θ : CP 1

u×CP 1z → CP 1

C×CP 1z

sending (u, z) to (C, z) where z 7→ z and

C =α

d− α· 1

u− z⇐⇒ u = z +

α

d− αC−1. (4.12)

On the subset z 6= u, Θ is an isomorphism with the complement to z = ∞. Therestriction Θ : D → C provides a birational transformation between D and thesymbol curve C determined by (1.7). Therefore D can be studied by using theresults about C from Section 3. We will need the following properties.

Lemma 9. Under the conditions imposed on P in Section 3, the following holds:

(i) D is an irreducible rational and smooth curve, and Θ extends to a mapD → C which is a resolution of singularities of C.

(ii) D has d branches over a neighbourhood of z = ∞. One branch passesthrough (∞,∞). The remaining branches pass through d − 1 points of theform (qi,∞), where q1, . . . , qd−1 are d− 1 (distinct) roots of P ′(u) = 0.

(iii) There are no points with u =∞ on D, except (∞,∞). The intersection ofD with the diagonal u = z consists of the points (pi, pi), i = 1, . . . , d, forwhich R(pi) = 0, together with (∞,∞).

(iv) The meromorphic form C dz on D given by

C dz :=α

d− αdz

u− z(4.13)

has its (simple) poles at the points:(a)(u, z) = (pi, pi), i = 1, . . . , d, with residue (1− α)/(d− α),(b) (u, z) = (qi,∞), i = 1, . . . , d− 1, with residue α/(d− α),(c) (∞,∞) with residue −1.

Proof. To settle (i)-(iii), notice that for z finite, we have that u = ∞ ⇐⇒ C = 0by (4.12), and the last equality does not happen for finite z, (comp. the proof ofTheorem 8 (iv)). The local coordinate around z = ∞ is y = 1/z and then (4.9)becomes

P ′(u)(yu− 1) = αP (u)y. (4.14)

At y = 0, (4.14) simplifies to P ′(u) = 0, proving that there are exactly d− 1 points(qi,∞), i = 1, . . . , d − 1, with u = qi finite. Since P (u) has only simple zeros, thegradient to (4.14) at y = 0, u = qi equals

(−P ′′(qi), αP (qi)) 6= (0, 0),

proving that the branches of D passing through (∞, qi) are smooth. Similar calcu-lation gives that the branch of D passing through (∞,∞) is smooth as well. In the(u, z)-plane, the gradient of P ′(u)(u− z)− αP (u) is given by

(P ′′(u)(u− z) + P ′(u)− αP ′(u),−P ′(u)) 6= (0, 0).

Non-vanishing of this gradient is the consequence of the fact that the conditionP (u) = P ′(u) = 0 contradicts to our assumption that P has only simple roots.Hence D is a smooth curve, and Θ : D → C extends to a resolution of singularitiesof C (by sending d points of D on the line z = ∞ to (0,∞), and another d points


(pi, pi) to (∞, pi)). Note, in particular, that the singularity of C at (0,∞) is resolved.D is clearly rational and irreducible, since C has these properties, see Theorem 8i). The remaining statements follow from similar calculations.

To settle (iv), note that Θ(pi, pi) = (∞, pi), and that the d branches of C passingthrough (0,∞) correspond to the d branches of D which intersect z =∞. Hence thecalculations of the residues on C in Theorem 8, (v) – (vi) apply. The coordinatesWand C are related by C = (α/(d− α)) dW. Therefore, to get the residues of C dzin (4.13), one should multiply the residues ofW dz in Theorem 8 by α/(d−α).

4.0.4. G(w, z0) distinguishes maximal saddle points almost everywhere. For a fixedz0 which is not a zero of P (w), let w1, w2, . . . , wd be the saddle points of (4.1).Assume that w1 is maximal, i.e. it satisfies the condition

G(w1, z0) > G(wj , z0), j = 2, . . . , d. (4.15)

Lemma 10. There exists an open dense subset U ⊂ Cz such that, for every z ∈ U ,there exists a unique maximal saddle point w1.

Proof. We argue by contradiction. Suppose that D ⊂ Cz is an arbitrary diskcontaining no branch points of the curve D; therefore there exist d branches of Ddefined in D which we denote by wi(z), i = 1, . . . , d. In a dense set two harmonicfunctions are either equal or distinct. VAD MENAS HR? Hence we can assume thatG(w1, z0) = G(wj , z0), in the disk. By analytic continuation, using the irreducibilityof D (Lemma 9 i)), the branch representing w1 can be analytically continued to anyother branch. In particular, it can be continued to the one for which u→∞. But forthe corresponding continuation of the other branch, we have u→ qi ∈ C, for someqi : R′(qi) = 0 (by Lemma 9 ii)). Substituting these values into G(u, z0) gives acontradiction to the equality. I CAN NOT UNDERSTAND! Namely, for the secondbranch G(u, z) ∼ − log |z|, and for the first branch G(u, z) ∼ ((d−α)/α) log |z|, bya calculation using Theorem 8 iii) to get that log |u| ∼ − log |z|.

4.0.5. A contour passing through the maximal saddle point. Assuming the validityof (4.15), we now show that it is possible to deform the original integration contourc from (4.1) to a contour γ posessing the following two properties:

(*) the curve γ = ∆ + γ1 passes through w1 along a curve ∆ of steepest de-scent/ascent of the function G(w, z0);(**) there exist δ > 0 such that, for all points w ∈ γ1, G(w, z0) < G(w1, z0)− δ.

For z0 fixed, consider the graph Z of w 7→ G(w, z0), as a mountainscape. Clearlyall the critical points of G(w, z0) are the saddle points, since a harmonic functioncan not have local minima or maxima. At the pole w = z0, G(w, z0)→ +∞; whileat all other poles w (which are given by P (w) = 0 one has that G(w, z0) → −∞).In addition, G(w, z0)→ +∞ when |w| → ∞.

Intuitively, we obtain a contour satisfying the above conditions by taking a curvec above a small circle c around z0 on the graph Z. Push c down on Z, away fromthe peak at z0, until it hits the saddles in the mountainscape. Then, additionallydeform c locally at the maximal saddle (w1, G(w1, z0)), so that it passes throughthis point along a path of steepest descent/ascent, ensuring that the projection ofthe deformed curve in the w-plane satisfies (**). Details are as follows.

Set δ0 := minG(w1, z0) − G(wj , z0), j = 2, . . . , d. Let N be a small neigh-

bourhood of w1, and let ∆ be a path of steepest descent/ascent through w1 in N .


Choose points a and b on ∆, on each side of w1, such that

G(a, z0) = G(b, z0) = G(w1, z0, )− δ > G(w1, z0, )δ0.

The curve ∆ in (**) will be the segment of ∆ between a and b.

Now take Ξ to be any piecewise C1-path connecting a to b which (i) encircles z0

once and (ii) avoids N . Set d(Ξ) := maxG(w, z0) | w ∈ Ξ. If p ∈ Ξ is not a saddlepoint, then there is a neighbourhood Np of p divided into two open subsets by thecurve G(q, z0) = G(p, z0); one of these subsets is N−p = q : G(q, z0) < G(p, z0).

If d(Ξ) > G(w1, z0)− δ, then the compact set M of w such that G(w, z0) = d(Ξ)consists of a finite number of points (since G is real-analytic) and possibly a finite

number of segments of Ξ, and does not contain a or b. Let p be an isolated pointin M . Since

G(p, z0, ) ≥ G(w1, z0)− δ > G(wj , z0), j = 2, . . . , d

p is not a saddle point. Furthermore, we can deform the part of Ξ contained ina small neighbourhood Np of p, so that it fits in N−p , and hence G(w, z0, w) <d(Ξ ∩ Np). For a segment the argument is similar. Using compactness, we can

repeat the construction for the whole M and get a curve Ξ1 such that d(Ξ1) < d(Ξ).Hence infΞ d(Ξ) = G(w1, z0) − δ. In particular, we can find a curve Ξ such that

d(Ξ) = G(w1, z0)− δ/2. Set δ = δ/2, and let γ1 be Ξ. This proves the claim.The latter argument implies the convergence of our Cauchy integral almost ev-

erywhere which, in its turn, implies that the assumption in the following corollaryis satisfied for z0 in a dense open subset, see Lemma 10.

Corollary 4. Assume that (w1, z0) ∈ D is such that

G(w1, z0) > G(wj , z0), j = 2, . . . , d. (4.16)

Then,

limm→∞

|I(m, sn, c)|1/m = eG(z0,w1), (4.17)

where c is any contour encircling z0 once counterclockwise.

Proof. Deform c as in the first part of this section. If M is the upper bound of|P sn(w)| on γ1, then by (**)

|I(m, sn, γ1)| ≤∫γ1

eG(w,z0)m|P sn(w)| dz ≤M Length(γ1)em(G(w1,z0)−δ). (4.18)

Comparison of (4.18) with the limit of the integral on ∆ given by (4.8) provesthe corollary.

4.0.6. Convergence of the logarithmic potentials almost everywhere. By (4.1), thevalue of the monic polynomial qn proportional to qn at z0 is given by

qn(z0) :=(nd− ([αn]− 1))!

(nd)!qn(z0) =

([αn]− 1)!(nd− ([αn]− 1))!

2πi(nd)!

∫c

Pn(z) dz

(z − z0)[αn].

(4.19)The degree of qn equals dn = nd − (m − 1), where m = [αn]. Recall that the

logarithmic potential Lµn(z) of the root-counting measure µn of qn can be expressedas

Lµn(z) = log |qn|1/dn .


By (4.2),

n =m

α+ sn,

where 0 ≤ sn < 1/α. Hence

dn =

(d− αα

)m+ (snd+ 1) = βm+O(1), (4.20)

with β := d−αα .

Lemma 11. In the above notation,

limn→∞

1

dnlog

((m− 1)!(nd− (m− 1))!

(nd)!

)=β log β − (β + 1) log(β + 1)

β:= B.

Proof. Straight-forward calculation using Stirling’s formula.

Now we can calculate the limit of the sequence (1/dn) log |qn| of logarithmicpotentials by taking the logarithm of (4.19) and using Lemma 11 together with(4.17).

Corollary 5. For any point z0 such that there is a unique maximal saddle pointw1, one has

limn→∞

Lµn(z0) = B +1

αβ(log |P (w1)| − α log |w1 − z0|).

4.0.7. Convergence in L1loc and final steps in the proof of Theorem 1. The next

theorem is our strongest convergence result important for the rest of the paper. Set

H(u, z) := β−1G(u, z) :=1

d− α(log |P (u)| − α log |u− z|).

Theorem 12. For any polynomial P of degree d ≥ 2,

limn→∞

Lµn(z) = B + π∗H(u, z),

where the limit is understood in the L1loc-sense and π∗ denotes the pushforward

induced by the projection πz : Cz × Cu → Cz. Consequently,

limn→∞

Cµn(z) = 2∂π∗H(u, z)

∂z,

and,

limn→∞

µn = µ :=2

π

∂2π∗H(u, z)

∂z∂z,

where the last two limits are understood in the sense of distributions.

To settle Theorem 12, we need the following useful corollary of Vitali’s conver-gence theorem. It provides a criterion for L1

loc-convergence under the assumptionthat one has the pointwise convergence almost everywhere.

Lemma 13. Let pn be a sequence of monic polynomials of strictly increasing

degrees dn := deg pn → ∞ as n → ∞. Denote by µn := 1dn

∑dni=1 δζi the root-

counting measure of pn, where ζi, i = 1, . . . , dn are the zeros of pn. Denote byLn(z) := 1

dnlog |pn(z)| the logarithmic potential of µn. Assume that

(i) there is a compact set K ⊂ C containing all the zeros of pn, n = 1, . . . ;(ii) the sequence Ln(z) converges to L(z) pointwise a.e. in C.

Then, L is a L1loc-function and limn→∞ Ln = L in the L1

loc-sense.


Proof. By Vitali’s theorem, we only need to check the uniform integrability of ourfunctions on an arbitrary fixed compact set M ⊃ K. Let E be a set with Lebesguemeasure λ(E) < ε < 1. Introduce

log+(x) := | log |z|| = f<ε(x) + f≥ε(x),

where f<ε(x) = log+(x) = − log |x|, if 0 < x ≤ ε and f<ε(x) = 0, if x > ε. (Thusf≥ε(x) = log+(x), if x > ε and 0 if 0 < x ≤ ε.)

We obtain∣∣∣∣∫E

Lµn(z − ζ) dλ(ζ)

∣∣∣∣ ≤ 1

dn

dn∑i=1

∫E

log+(z − ζi) dλ(ζ) (4.21)

≤ 1

dn

dn∑i=1

∫E

f<ε(z − ζi) dλ(ζ) +1

dn

dn∑i=1

∫E

f≥ε(z − ζi) dλ(ζ) := I1 + I2. (4.22)

If Dε(ζi) is a disk of radius ε centered at ζi, then∫Dε(ζi)

| log |z − ζi|| dz = −πε2(log ε− 1/2). (4.23)

Hence ∫E

f<ε(z − ζi) dλ(ζ) ≤∫Dε(ζi)

f<ε(z − ζi) dλ(ζ) = −πε2(log ε− 1/2),

implying

I1 ≤−1

dn(dnπε

2(log ε− 1/2)) = O(ε)

with a constant which only depends on ε. Let δ be the diameter of M . For thesecond sum above, let m := max− log ε, log+(δ) be an upper bound of f≥ε(x− ζ)for x, ζ ∈ M . Then I2 ≤ mλ(E) ≤ mε = o(1) as ε → 0. This proves that Ln(z)are uniformly integrable. Lemma 13 is now immediate from Vitali’s convergencetheorem.

Proof of Theorem 12. Corollary 5 says that we have the desired pointwise conver-gence a.e. in C, which together with Lemma 13 implies the first part of Theorem12. The other parts follow from some basic properties of distributions, since L1

loc-convergence implies convergence as distributions. Notice that µ and its Cauchytransform are distributional derivatives of the logarithmic potential of µ.

We now finalize our proof of Theorem 1. By Theorem 12 (cf. Lemma 10) C iscovered a.e. by open sets Ui, i ∈ I such that in each Ui there is a branch u = u(z)of D, so that the equality:

H(u(z), z) +B = Lµ(z)

holds. Consequently, in each Ui

∂β−1G(u, z))

∂z= Cµ(z),

and u = z + (βCµ)(z)−1. The equation defining D satisfied by u implies that theCauchy transform C = Cµ of µ satisfies, a.e. in C, the equation

(d− α)C =d(logP

(z + (βC)−1

))dz

. (4.24)


Formula (4.24) coincides with the equation (1.3) which settles Theorem 1, up to asmall shift of the order of derivation. We have actually proven that the sequence ofroot-counting measures for (Pn)[αn]−1 converges, but using e.g. the main resultof [To], we get that the sequence considered in Theorem 1 has the same limit asthat of (Pn)[αn].

Proof of Corollary 3. Follows immediately from Theorem 12, using the descriptionof the tropical trace in Theorem 4 as a piecewise harmonic function and takinginto account the description of the poles of H(w, z) on the saddle point curve, inLemma 9 (iv).

4.0.8. The saddle point curve D and another proof of the fact that C is an aBc.Recall the general construction of affine Boutroux curves given in Section 2.5.1. Aswe will see now, the curve C is a particular instance of this construction.

The starting point is the pluriharmonic function

H(u, z) := β−1G(u, z) :=1

d− α(log |P (u)| − α log |u− z|).

It is well-defined for all (u, z) ∈ C2 except for those where either P (u) = 0 or u = z.Its differential is a meromorphic 1-form given by

d(H(u, z)) :=1

2(d− α)

((P ′(u)

P (u)− α

u− z

)du+

α

u− zdz

).

D is the rational plane curve given by

2(d− α)∂H

∂u=P ′(u)

P (u)− α

u− z= 0.

Restricting H to D, we get a simplified expression for its differential given by:

dH(u, z) =1

2β

1

u− zdz, (u, z) ∈ D.

Consider the projection πz : D → Cz sending (u, z) to z. Except for a finitenumber of branch points, z is a local coordinate on D. Since D is smooth byLemma 9, then in some neighbourhood of any point p = (u, z) ∈ D, the restrictionof H to D is a real-valued harmonic function satisfying

H(p)−H(p0) = Re

∫ p

p0

1

β(u− z)dz,

where p0 is another fixed point on D. In particular, this implies that the form

ω = (β(u− z))−1 dz

has imaginary periods on D which also follows from Proposition 8. Using the changeof coordinates

v =1

β(u− z), z = z ⇐⇒ u = z + (βv)−1, z = z,

we see that D considered as a plane curve in coordinates (v, z) is an affine Boutrouxcurve, since ω = v dz. The tropical trace of H(p) under the projection πz : D →

CP 1 is given by

π∗H(u, z) := maxp∈DH(p)|πz(p) = z.


Note that this trace does not depend on the particular choice of coordinates in C2

used to describe D as a plane curve; it only depends on the projection πz and the

function H. In terms of the coordinates (v, z), H(u, z) = H(v, z), where

H(v, z) = β−1G(u, z) :=1

αβ(log |P (z + (βv)−1)|+ α log |(βv)|),

and π∗H(z) = π∗H(z), where π : (v, z) 7→ z.

Since dH = 12v dz, this gives an alternative proof of Theorem 8 claiming that C

is an aBc.

5. Differential equations satisfied by Rodriques’ descendants

In this and the following sections we will prove our main results formulated inthe Introduction. First, we need to obtain linear differential equations satisfied bythe Rodriques’ descendants.

Having in mind future applications and generalisations, we will deduce lineardifferential equations satisfied by the Rodrigues’ descendants of the meromorphicfunctions of the form

f(z) := P (z)eT (z)/Q(z),

where P (z) 6≡ 0, Q(z) 6≡ 0 and T (z) are polynomials with gcd(P,Q) = 1. In caseT ≡ 0 and Q ≡ 1, we obtain the polynomial situation considered in the presentpaper, see Corollary 7.

Proposition 14. In the above notation and for d := degP + degQ + deg T , theRodrigues’ descendant Rm,n,PeT /Q(z) satisfies the linear homogeneous differentialequation

d∑i=0

i∑j=0

j∑k=0

(m+ d− i+ n(2j − i))δk,0 − nkT (k)

(m+ d− i)!(i− j)!(j − k)!k!P (i−j)Q(j−k)y(d−i) = 0 (5.1)

of order d. Here δk,0 =

1, if k = 0

0, otherwise.

As special cases of the latter statement we obtain the following three corollaries.

Corollary 6. The Rodrigues’ descendant Rm,n,P/Q(z) satisfies the linear homoge-neous differential equation

d∑i=0

i∑j=0

m+ d+ (n− 1)i− 2nj

(m+ d− i)! (i− j)! j!P (j)Q(i−j)y(d−i) = 0 (5.2)

of order d = degP + degQ.

Corollary 7. The Rodrigues’ descendant Rm,n,P (z) satisfies the linear homoge-neous differential equation

d∑i=0

(m− nd)− (i− d)(n+ 1)

(d+m− i)! i!P (i)y(d−i) = 0 (5.3)

of order d = degP .

The original Rodrigues’ formula inspires the following consequence of Corollary 7.


Corollary 8. The Rodrigues’ descendant Rn,n,P (z) := dn

dzn (Pn(z)) satisfies thelinear differential equation

d∑i=0

(d− 1)− (i− 1)(n+ 1)

(d+ n− i)! i!P (i)y(d−i) = 0 (5.4)

of order d = degP .

Remark 10. A differential equation satisfied by Rn,n,P (z) similar to (5.4) wasorginally obtained in [Ho].

Proof of Proposition 14. Consider the first-order differential equation

PQw′ + n(PQ′ − P ′Q− PQT ′)w = 0, (5.5)

Clearly, if f = PeT /Q, then w = fn satisfies (5.5). By differentiating both sidesof (5.5) ` ≥ d − 1 times (or ` > d − 1 times if d = 0) and using Leibniz’s rule forthe derivative of a product, we get

∑i=0

(`

i

)U (i)w(`+1−i) + n ·

∑i=0

(`

i

)V (i)w(`−i) − n ·

∑i=0

(`

i

)W (i)w(`−i) = 0, (5.6)

where U := PQ, V := PQ′ − P ′Q and W := PQT ′. In the first sum, remove thefirst term and replace i by r + 1 in the remaining sum. In the second and thirdsums, replace i by r and remove the last terms. By combining the three resultingsums and simplifying, equation (5.6) becomes

Uw(`+1)+nV (`)w−nW (`)w+

`−1∑r=0

(`

r

)(`− rr + 1

U (r+1) + nV (r) − nW (r)

)w(`−r) = 0.

(5.7)By changing the upper limit of summation in (5.7) from `−1 to `, the terms nV (`)wand −nW (`)w are encompassed by the sum. Since U, V and W are polynomialsof degrees at most d, d − 1 and d − 1, respectively (where we let deg 0 := 0), and` ≥ d−1, we can change the upper limit of summation further to d−1, since higherterms vanish. That is, we obtain the equation

Uw(`+1) +

d−1∑r=0

(`

r

)(`− rr + 1

U (r+1) + nV (r) − nW (r)

)w(`−r) = 0,

or equivalently, if we replace r by i − 1, change the lower index of summation toi = 0, and define 0 · V (−1) = 0 ·W (−1) = 0 as to not introduce any new terms,

d∑i=0

`!

(`− i+ 1)! i!

((`− i+ 1)U (i) + niV (i−1) − niW (i−1)

)w(`−i+1) = 0. (5.8)

Since the terms in U and V contain two factors, while W contains three factors,we expand their derivatives using Leibniz’s rule as follows:

U (i) = (P ·Q · 1)(i) = PQ(i) +

i−1∑j=0

j∑k=0

(i

i− j, j − k, k

)P (i−j)Q(j−k)δk,0, (5.9)


V (i−1) = (P ·Q′ · 1)(i−1) − (P ′ ·Q · 1)(i−1)

=

i−1∑j=0

j∑k=0

(i− 1

i− j − 1, j − k, k

)(P (i−j−1)Q(j−k+1) − P (i−j)Q(j−k)

)δk,0,

(5.10)

W (i−1) = (P ·Q · T ′)(i−1) =

i−1∑j=0

j∑k=0

(i− 1

i− j − 1, j − k, k

)P (i−j−1)Q(j−k)T (k+1).

(5.11)By inserting the expressions in (5.9)-(5.11) into (5.8) and simplifying, we see that

d∑i=0

(`

i

)PQ(i)w(`−i+1) +

d∑i=0

i−1∑j=0

j∑k=0

`!

(`− i+ 1)!(i− j)!(j − k)!k!

[(`− i+ 1)P (i−j)Q(j−k)δk,0

+ n(i− j)((P (i−j−1)Q(j−k+1) − P (i−j)Q(j−k)

)δk,0 − P (i−j−1)Q(j−k)T (k+1)

)]w(`−i+1) = 0.

(5.12)

By changing the upper index of summation from i − 1 to i in (5.12), and usingthe convention that 0 · P (−1) = 0 as previously, the first sum is encompassed bythe triple sum. Next, reverse the order of summation in the outer sum, and letm := `− d+ 1, which gives that

d∑i=0

d−i∑j=0

j∑k=0

(m+ d− 1)!

(m+ i)!(d− i− j)!(j − k)!k!

[(m+ i)P (d−i−j)Q(j−k)δk,0 + n(d− i− j)×((

P (d−i−j−1)Q(j−k+1) − P (d−i−j)Q(j−k))δk,0 − P (d−i−j−1)Q(j−k)T (k+1)

)]w(m+i) = 0,

(5.13)

for all ` = m + d − 1 ≥ d − 1 ⇐⇒ m ≥ 0. Now let (∗) denote the equationobtained by replacing w(m+i) by y(i) in (5.13). Clearly, y = w(m) = (fn)(m) =((PeT /Q)n)(m) satisfies (∗). Thus, by reversing the order of summation in (∗) andsimplifying, the proposition follows.

Corollaries 6, 7, and 8 are immediate consequences of Proposition 14.DESRCRIBE THAT THE SADLLE POINT CURVE CAN BE OBTAINED

FROM THE ABOVE DIFFERENTIAL EQUATIONS!

6. Example of a quadratic polynomial P (z)

A very basic example of the considered situation is related to Legendre poly-nomials and the original Rodrigues’ formula in which case we will exemplify ourresults explicitly. Then P (z) = z2−1 and for the sequence of Legendre polynomials(1.1), the density of their asymptotic root distribution equals

ρ(x) =2

π

√1− x2dx, x ∈ [−1, 1].

More generally, for a given 0 < α < 2, considering the sequence

R(α)n (x) :=

d[αn](x2 − 1)n

dx[αn],


the smooth rational plane algebraic curve D that plays an important role in ourcalculations is defined by the equation

P ′(u)(u−z) = αP (u) ⇐⇒ (2−α)u2−2uz+α = 0 ⇐⇒ u =1

2− α(z±

√z2 + α2 − 2α).

It has branch points corresponding to z = ±√α(2− α). Also the pluriharmonic

function H is

H(u, z) =1

2− α(log |u2 − 1| − α log |u− z|

)=

1

2− α

(log

(2

α

)+ log |u|+ (1− α) log |u− z|

).

++++++ We get that for 2 > α ≥ 1 the asymptotic root distribution will besupported on the interval Iα = [−

√α(2− α),

√α(2− α)] with the density given

by

ρ(α)(x) =2

π

√2α− α2 − x2, x ∈ Iα,

while for 1 > α > 0 the asymptotic root distribution will also contain point massesat ±1. Note that Ia is bounded by the branch points.

Hence we may intuitively describe what happens when α increases from 0 to 2,as follows. Part of the mass of the two point measures move out of ±1, first tothe midpoint, then expanding to a measure supported on the interval Ia aroundthis point, until the measure has support on the whole interval at α = 1. Thenwhen 1 < a < 2 there is no mass left att ±1 and the support contracts again to themidpoint of the interval (and vanishes at α = 2). See figure ??. In particular, forall 0 < α < 2, Iα is contained in [−1, 1] and is strictly smaller than [−1, 1] for allα 6= 1

ρ(α)(x) =2

π

√2α− α2 − x2, x ∈ Iα.

.

7. Final Remarks and open problems

1. Practically all the results of the present paper can be generalised to the casewhen f is a rational function instead of a polynomial which we plan to do in thefuture. However poles of a rational function restrict the possibility of deformationof the integration contour used in § 5. This leads to a more delicate situation whichrequires special analysis.

2. The set-up of the present paper can be randomised and generalized as fol-lows. Let ξ be a probability measure compactly supported in C. Denote byPn =

∏ni=1(x− ξi) a random polynomial of degree n whose roots are i.i.d. random

variables sampled on ξ. Given a sequence A = αn of non-negative integers, set

Qn = P(αn)n and denote by µn the root-counting measure of Qn. Results from the

recent papers [PeRi, Ka] motivate the following guess.

Conjecture 1. In the above notation, the following two statements hold:

(i) if αnn → 0, then the sequence µn converges in probability to ξ;

(ii) if αnn → α, 0 < α < 1, then the sequence µn converges in probability to a

measure ξα whose support is contained in the convex hull of the support of ξ;


What we have done in the present paper can be interpreted in the above termsas follows. We start with a uniform discrete measure ξ supported on the d zerosof P (z). Then we sample this measure evenly and deterministically nd times, byforming the series of polynomials Pn(z) := Pn(z) and, finally, we differentiate Pn(z)[nα] times. This produces a sequence of polynomials Qn(z) and the associatedsequence of probability measures µn. The proportion between the number ofderivations and the number of sampled points is

A :=α

dn+O(1).

Observe now that (log |P (z)|)/d is the logarithmic potential Lξ(z), and henceour main result Theorem 12 can be reformulated as

µn →2

π

∂2

∂z∂z

(1

1−Aπ∗Lξ(z + w−1)− A

1−Alog |w|

), (7.1)

where π∗Lξ(z +w−1) is taken as the maximum for the saddle point locus W of allw, such that

∂G(w, z)

∂w= 0,

where

G(w, z) :=1

1−ALξ(w)− A

1−ALog |w − z|.

Such interpretation of Theorem 12 makes sense if ξ is an arbitrary probabilitymeasure such that polynomials Qn(z) are obtained by sampling independentroots according to ξ. It seems plausible that the relation (7.1) holds for much moregeneral measures than special measures with finite discrete support which we haveconsidered in this paper.

3. Numerical experiments similar to the one shown in Fig. 3 motivate the followingguess.

Conjecture 2. For any polynomial P of degree at least 2,(i) the union of the supports ΥP = ∪α∈[0,d]Sα,P is a domain inside the convex hullof the roots of P . The boundary of ΥP is a subset of the curve formed by the familyof branch points of Γα,P depending on α;(ii) ΥP is a concave domain.

4. Let P be a cubic polynomial with noncollinear zeros, and let h = h(W )denote the left-hand side of equation (1.5). If we solve the quartic equationResultant(h, h′)/P = 0 in z for α = 0, 1, 3, the twelve solutions that arise arevarious triangle centers associated with the triangle defined by Z(P ). These are,not counting multiplicity, its center of mass, Z(P ), Z(P ′), its first and second iso-dynamic points (denoted I1 and I2, respectively), and an additional point that wedenote A. Numerical experiments indicate that the (complementary) support ofZ(Rn,n,P ), n → ∞, has non-obvious connections to some of these points, and iseither a tree with three edges that have a vertex in common, or two disjoint edges;see Fig. 4.

5. We conjecture that some of the local behavior of the zeros of Rn,n,P can becharacterized in terms of the zeros of the n-th Legendre polynomial Ln as follows:


0 1 2 3 4 5 6Re

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Im

0 2 4 6 8Re0.0

0.5

1.0

1.5

2.0

Im

Figure 4. The zeros ofR100,100,P (z) (shown by small dots). Here,

the square is the branch point A, and the two lines are AI1 andAI2, respectively. Note that one of the two lines consistently seemsto pass through the vertex of degree 3 in the tree when it exists.

Conjecture 3. Let P := (z − 1)(z + 1)∏dj=1(z − καj) be a polynomial, where d ∈

N, κ ∈ R, and αj ∈ C\0, j = 1, . . . , d. Then, for any n ∈ N, Z(Ln) ⊆ Z(Rn,n,P )when κ→∞.

Additionally, we sketch a rough explanation of some of the apparent global be-havior of the zeros. Let P be a polynomial with all of its d1 + d2 + . . . + drzeros clustered into non-overlapping disks D1, D2, . . . , Dr centered at z1, z2, . . . , zr,respectively, where smaller disks (in some sense) will lead to better approxima-tions. Furthermore, let µP denote the zero-counting measure of P (z), and letU := C \ D1 ∪ . . .∪Dr. Then we conjecture that supp(µR)∩U with R = Rn,n,Pcan be approximated by supp(µR) ∩ U with R = Rn,n,W as n → ∞, where

W := (z− z1)d1(z− z2)d2 · · · (z− zr)dr . Here, the error in the approximation seemsto be inversely proportional to the distances between the clusters (and disks), sincelarger distances intuitively lead to more concentrated (and, thus, more pointlike)masses per cluster, as seen by an observer above C who maintains a fixed viewingangle between the outermost zeros of P .

6. Our final remark concerns Theorem 7.

Conjecture 4. Under the assumptions of Theorem 7 the signed measure whoseexistence is claimed in this result is unique.

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Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

E-mail address: [email protected]

Department of Mathematics, Stockholm University, SE-106 91 Stockholm, SwedenE-mail address: [email protected]

Department of Mathematics, Stockholm University, SE-106 91 Stockholm, SwedenE-mail address: [email protected]

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