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Catching all those vagrant roots

Marco Bertola, Concordia Univ. and C.R.M.

November 9, 2007

Abstract The roots we are after are those of families ofpolynomials that satisfy orthogonality relations of somesort. For orthogonal polynomials w.r.t a positive measureon the real line the problem is at least one century old, butnew results appear daily. Classical theory establishes thatall the zeroes are real and simple and in the limit of largedegrees they form some nice distribution. The reasonof the interest is the relation of orthogonal polynomials torandom-matrix theory, integrable systems, analysis, geom-etry.If we replace the positive measure by a complex-valuedanalytic weight then the zeroes are no longer real, butthey lie in the complex plane: yet, in the large-degree limitthey arrange themselves in nice patterns (trees of Jordanarcs typically).So the question arises: how do we find all those flyingzeroes? Is there something to say about the geometry ofthese trees?I will touch upon these issues, showing some nice picturesand animations, and also on some recent conjectures onthe zeroes of orthogonal polynomials for area-integrals inthe plane

Marco Bertola, Concordia Univ. and C.R.M. Vagrant Roots

A similar old problem OPs What to ask Hermite The density Going complex Show Time OPs in the plane

Taylor polynomials

Where are the zeroes of the Taylor polynomials of ez?

From Hans Lundmarks web-page:

http://www.mai.liu.se/ halun/complex/taylor/

ez Pn(z) =nX

j=0

1

j!z j

The exponential function has no zeroes! Pn(z)has n zeroes!. They fly away : upon rescaling

pn(z) := Pn(nz)

the zeroes of pn(z) become dense on the curve

:= {z : |ze1z | = 1 , |z| 1} (Szego, 1924)



Orthogonal polynomials

Given a positive measure (x)dx on R, O.P.s are a sequence of monic polynomialspn(x) = xn + . . . such that Z

Rpn(x)pm(x)(x)dx = hnnm

They are Pade approximantsZR

(s)ds

x s

AKWeyl function

=qn(x)

pn(x)+O(x2n1) , qn(x) =

ZR

pn(x) pn(s)x s

(s)ds

Pade approximants are to the Weyl functions what continued fractions are to realnumbersThey enter in an approximation scheme similarly to Taylor polynomials.



What are the interesting questions

Where are the zeroes of OPs? More precisely, how to describe the (normalized)counting measure of zeroes

n =1

n

nXj=1

x

(n)j

, pn(x(n)j ) = 0

Can we describe limn pn(x)? Yes: we need however advanced analysis like

RiemannHilbert problems;algebraic geometry of hyperelliptic curves;functions and half-differentials.

Known properties (Szego)

The zeroes of pn(x) are all real, simple and in the convex hull of supp().

The zeroes of pn(x) and pn+1(x) are interlaced.



An example: Hermite polynomials

ZR

ex2Hn(x)Hm(x) = mn2

nn!

n

then (PlancherelRotach) for x =

2n + 1 cos

ex2/2Hn =

2n/2+14 (n!)

12

(n)14 (sin)

12

sin

n

2+

1

4

(sin 2 2) +

3

4

+O(n1)

Yucky!But we can read off the (scaled) position of the zeroes and see that their distributionis approaching the Wigner semicircle law

() =1

p2 2 , =

x

n



The density of zeroes

n

T=

1

T > 1

= nT

T 0

Then the OPs depend also on = nT

(i.e. T = n

)

pn(x ; ) =nY

j=1

(x x(n)j ) n, =1

n

nXj=1

x

(n)j

Theorem (a long list of people.... Pastur, Deift, Johansson,....)

The density n, -= n

T, n

1

T(x) where is the equilibrium density of a distribution

of charges on the real axis,R(x) = T, subject to the external electrostatic potential

V (x).



Stepping out of reality: pseudo-OPs

Pseudo Orthogonal polynomials are monic polynomials pn(x) = xn + . . . such thatZpn(x)pm(x)e

V (x)dx = hnnm

where V (x) is COMPLEX! (and also the contour of integration may be in thecomplex plane).Q: where are the zeroes going now? (i.e. where are the poles of the Pade approximants)

Theorem

Suppose V (x) is a complex polynomial. Then n, where is a positive measureof mass T supported on a tree of Jordan arcs. This tree grows as T grows.

Finding the tree is a problem in harmonic analysis (harmonic function + BVconditions + free boundary) - Boutroux curves



Some nice animations

Example: V (x) = x4 + x2 and Tgrowing. Different evolutions cor-respond to different choices of con-tours of integration. The real axisis not an option even if V is real

since ex4+x2 is not integrable on

the real axis.


CA.aviMedia File (video/avi)

CB.aviMedia File (video/avi)

CD.aviMedia File (video/avi)

CE.aviMedia File (video/avi)


Another growing tree ...

(Three hours on the four CPUs of Cuma....)


Nicetree.aviMedia File (video/avi)


...and another ...

(Five hours on the four CPUs of Cuma....)


Nicetree2.aviMedia File (video/avi)


... and in full glory!

(A night on the four CPUs of Cuma....)


Nicetree3.aviMedia File (video/avi)


Other (static) examples of nice trees

Three possible solution of cuts and hyperelliptic Boutroux curves for the potential

V (x) = x6

6and total charge T = 1.

V (x) = x8

4+ ix

6

6 x

2

2+ 3x , T = 10. V (x) =

x7

7+ ix

5

5 x

3

3+ x

2

10, T = 10.



This surface representsthe effective electrostaticpotential of V (x) = x6

plus the electrostatic po-tential generated by (x),supported on the creases.



Orthogonal Polynomias in the plane

A different situation here: consider holomorphicpolynomials pn(z) = zn + . . . such thatZZ

Cpn(z)pm(z)e

(|z|2+H(z))d2z = hnnm

Where are the zeroes now?For example for the measure

e|z|2|z 1|2d2z

the zeroes of the OPs are in the animation onthe right: = 40 and n = 1, 2, . . . , (T = n

).

There are some very natural (numerically supported) conjectures.


Bean.aviMedia File (video/avi)


The region drawn around the zeroes (and thearc) are not fitted: they are analytical curves.The region describes (also) the (slow) growthof a bubble of air in a viscous fluid (in a 2Dgeometry) (Laplacian growth): the arc is theso-called skeleton (or mother-body) of the gen-eralized quadrature domain (the smiling bean).There is a connection to twodimensional fluiddinamics (zero-surface-tension, Darcyan). It isan open problem to prove what we plainly see....


Bean.aviMedia File (video/avi)


If we run time long enough a transition occurs:the domain closes on itself (the drawn domain isnot accurate then). These transitions are inter-esting and can be studied in their own sake inthe context of fluid-dynamics (see S. Y. Lee etal.)


Beanlong.aviMedia File (video/avi)


Fidos bones skeleton

3

2

1

0

1

2

3

1.510.5 0 0.5 1 1.5

For example for the measure

eN|z|2|z a|2N |z + a|2Nd2z

thezeroes of the OPs are in the picture on the right:N = 22 = n (T = 1) and a = 1.2588808....



Conclusions

There are several connections to other areas of mathematics/physics I did not mention

Pade approximations - Number Theory

Orthogonal polynomials and Random matrix models

RiemannHilbert analysis (nonlinear steepest descent method); a fantasticallypowerful method to obtain asymptotic expansions.

Isomonodromic deformations and integrable systems.

Algebraic geometry and harmonic analysis.

String theory.

Coherent states.

Group theory and differential geometry

....



The punchline

It is fair to say that the topic has tentacular ramifications.


Main TalkA similar (old) problemOrthogonal polynomialsWhat to askHermiteThe densityGoing complexShow TimeOrthogonal Polynomial in the plane

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