EXPECTED NUMBER OF RANDOM REAL ROOTS
TURGAY BAYRAKTAR
A BSTRACT. In this note, we obtain asymptotic expected number of real zeros of random
polynomials of the form
n
X
fn (z) =
anj cnj z j
j=0
where anj are independent copies of a real random variable ζ of mean zero, variance
one satisfying E|ζ|2+δ < ∞ and cnj z j form an orthonormal basis with respect to inner
product induced by e−2nϕ(z) dz for some non-negative circular C 2 -weight function ϕ
which has super logarithmic growth at infinity. In the special case, we recover results
of Edelman & Kostlan and Tao & Vu on the expected number of real roots of Weyl
polynomials.
1. I NTRODUCTION
In this note we consider random univariate polynomials of the form
fnζ (z)
(1.1)
=
n
X
anj cnj z j
j=0
where cnj are deterministic constants and anj are independent copies of a real valued
non-degenerate random variable ζ of mean zero and variance one. We denote the
number of real zeros of fnζ by Nnζ . Therefore Nnζ : Pn → {0, . . . , n} defines a random
variable on the set of polynomials Pn of degree at most n. We write fn and Nn for
short when ζ = NR (0, 1) is the real Gaussian.
Study of number of real roots for Kac polynomials (i.e. cnj = 1 for all j and n) goes
back to Bloch and Pólya [BP32] where they considered the case ζ is the uniform distribution on the set {−1, 0, 1}. This problem has been considered by Littlewood & Offord
in a series of papers [LO43, LO45, LO48] for real Gaussian, Bernoulli and uniform distributions. In [Kac43] Kac established a remarkable formula for the expected number
of real zeros of Gaussian random polynomials
Z p
4 1 A(x)C(x) − B 2 (x)
dx
(1.2)
ENn =
π 0
A(x)
where
A(x) =
n
X
j=0
2j
x , B(x) =
n
X
j=1
jx
2j−1
, C(x) =
n
X
j 2 x2j−2
j=1
2000 Mathematics Subject Classification. 30C15, 60G99.
Key words and phrases. Expected number of real zeros, random polynomial.
1
EXPECTED NUMBER OF RANDOM REAL ROOTS
2
and (1.2) in turn implies that
2
+ o(1)) log n.
π
Later, Erdös and Turan [ET50] obtained more precise estimates. These results are
also generalized to Bernoulli distributions by Erdös & Offord [EO56] as well as to
distributions in the domain of attraction of the normal law by Ibragimov & Maslova
[IM71a, IM71b].
On the other hand, for models other than Kac ensembles, the behavior of Nn changes
considerably. In [EK95] Edelman & Kostlan gave a beautiful geometric argument in order to calculate expected number of real roots of random polynomials. The argument
in [EK95] applies in a quite general setting of random sums of the form
ENn = (
fn (z) =
n
X
aj Pjn (z)
j=0
where Pjn are entire functions that take real values on the real line (see also [Van15,
LPX15] for recent treatment of this problem). In particular, Edelman and Kostlan
proved that
√
q n
 n
for Elliptic polynomials (i.e. cnj =
)
j
q
ENn =
√
( 2 + o(1)) n for Weyl polynomials (i.e. cn = 1 ).
j
π
j!
More recently, Tao and Vu [TV15] established some local universality results concerning the correlation functions of the zeroes of random polynomials of the form
(1.1). In particular, the results of [TV15] generalized aforementioned ones for real
Gaussians to the setting where ζ is a random variable ζ such that E|ζ|2+ < ∞. Moreover, Tao and Vu also obtained some variance estimate on Nn which leads to a weak
law of large numbers for real zeros.
We remark that all the three models above (Kac, elliptic, flat) arise in the context
of orthogonal polynomials. In this direction, expected distribution of real zeros of
random linear combination of Legendre polynomials is studied by Das [Das71] in
which he proved that
n
ENn = √ + o(n).
3
Recently, Lubinsky et. al [LPX15, PX15, LPX16] generalized this result to the random
orthogonal polynomials with suitable weights supported on the real line.
In this note we study real roots of random orthogonal polynomials which are random linear combinations of orthogonal polynomials induced by a non-negative radially
symmetric (i.e. ϕ(z) = ϕ(|z|)) C 2 -weight function satisfying
(1.3)
ϕ(z) ≥ (1 + ) log |z| for |z| 1
for some fixed > 0. Then one can define associated equilibrium potential ϕe and
1
weighted equilibrium measure is defined by µe := 2π
∆ϕe which is supported on the
compact set Dϕ := {z ∈ C : ϕe (z) = ϕ(z)} (see §2 for details). The interior Bϕ := {z ∈
C : ∆ϕ(z) > 0} is often called as Bulk in the literature.
EXPECTED NUMBER OF RANDOM REAL ROOTS
3
We define a L2 -norm on the space Pn of polynomials of degree at most n by
Z
2
(1.4)
kfn kϕ :=
|fn (z)|2 e−2nϕ dz
C
where dz denotes the Lebesgue measure on C. In particular monomials {z j }nj=0 form
an orthogonal basis for Pn . We consider random polynomials of the form (1.1) where
anj are independent copies of a real random variable ζ of mean zero and variance one
and
Z
1
n
cj := ( |z|2j e−2nϕ(z) dz)− 2 .
C
Our main result (Theorem 1.1) establishes asymptotics of expected number ENnζ
of real roots of random polynomials with independent identically distributed (iid)
random coefficients satisfying mild moment conditions:
Theorem 1.1. Let ϕ : C → R be a non-negative radially symmetric weight function of
class C 2 satisfying (1.3) and ζ be a non-degenerate real random variable of mean zero
and variance one satisfying E|ζ|2+δ < ∞ for some δ > 0. Then expected number of real
zeros of random orthogonal polynomials fnζ (z) satisfies
r
Z
1
1
1
ζ
(1.5)
lim √ ENn =
∆ϕ(x)dx
n→∞
π Bϕ ∩R 2
n
In order to prove Theorem 1.1 for real Gaussian ζ = NR (0, 1), we utilize Edelman
and Kostlan’s [EK95] approach together with near and off diagonal Bergman kernel
asymptotics obtained in [Bay, Ber09]. Then the assertion of Theorem 1.1 is a consequence of Theorem 2.4 which indicates that with high probability (complex) zeros of
a random polynomial accumulate in the bulk. For non-Gaussian case, we adapt the replacement principle of [TV15] to our setting in order to prove that ENnζ has universal
property in the sense that under the mild assumptions limn→∞ √1n ENnζ independent of
the choice of ζ.
q
2
nj+1
which in turn gives rise to
In the special case, ϕ(z) = |z|2 we obtain cnj =
πj!
scaled Weyl polynomials and we recover results of Edelman & Kostlan [EK95] and
Tao & Vu [TV15]. In §4, we provide a more general family of circular weights and
calculate the asymptotic expected number of real zeros of the corresponding ensemble
of random orthogonal polynomials.
2. B ACKGROUND
Let ϕ : C → R be a C 2 function satisfying
(2.1)
ϕ(z) ≥ (1 + ) log |z| for |z| 1
for some > 0. One can define equilibrium potential
ϕe (z) := sup{u(z) : u ∈ SH(C) such that u(z) ≤ log+ |z|+O(1) and u(z) ≤ ϕ(z) for z ∈ C}.
Then ϕe ∈ SH(C) satisfying ϕe (z) = log+ |z| + O(1) as |z| → ∞ (see [ST97, §1]).
Throughout this note we assume that ϕ(z) is non-negative and radially symmetric (i.e.
ϕ(z) = ϕ(|z|) for z ∈ C).
EXPECTED NUMBER OF RANDOM REAL ROOTS
4
In this special setting there is at least one another way of obtaining ϕe as follows.
We may define
Φ:R→R
Φ(s) := ϕ(z)
where s = log |z| denotes logarithmic coordinate. Then it turns out that
(2.2)
Φe (s) := sup{u(s) : u is convex, k
du
k∞ ≤ 1 and u ≤ Φ on R}
ds
satisfies Φe (s) = ϕe (z) for s = log |z|. Here, the condition k du
k ≤ 1 ensures that the
ds ∞
upper envelope ϕe has logarithmic growth at infinity. In particular, the graph of ϕe is
the convex hull that of ϕ.
It follow from [Ber09] that the corresponding weighted equilibrium measure defined
by Laplacian of the equilibrium potential in the sense of distributions
µe :=
1
∆ϕe
2π
which is supported on the compact set
Dϕ = {z ∈ C : ϕe (z) = ϕ(z)}.
We remark that µe is the unique minimizer of the weighted energy functional
Z Z
Z
Iϕ (ν) := −
log |z − w|dν(z)dν(w) + 2 ϕ(z)dν(z)
over all probability measures supported
R on Dϕ (see [ST97, Theorem 3.1.3]). Following
[ST97], the constant Fϕ := I(µe ) − ϕdµe is called the modified Robin constant for ϕ.
The inner product
Z
hf, gi =
f (z)g(z)e−2nϕ(z) dz
C
2
induces an L -norm, denoted by k · kϕ , on the vector space of polynomials Pn of degree
at most n. Since ϕ is radially symmetric Pjn (z) = cnj z j form an orthonormal basis with
respect to this norm when cnj := (kz j kϕ )−1 . It’s well known that (cf. [ST97])
1
log |cnn | = Fϕ .
n
The next proposition is of independent interest and will be useful in the sequel. It
implies that the equilibrium potential can be obtained as upper envelope
(2.3)
lim
n→∞
ϕe (z) = (lim sup
j→∞
1
log |Pjn (z)|)∗
n
where Pjn (z) = cnj z j denote orthonormal polynomials.
Proposition 2.1. For every > 0 and z ∈ C there exists δ > 0 such that for sufficiently
large n
#{j ∈ {0, . . . , n} : cnj |z|j > en(ϕe (z)−3) } ≥ δn.
EXPECTED NUMBER OF RANDOM REAL ROOTS
5
R
Proof. We denote probability measures µn = a1n e−2nϕ(z) dz where an := C e−2nϕ(z) dz.
Since ϕ(z) is non-negative, it follows from Laplace’s principle that (cf. [DS89, §1])
{µn }∞
n=1 satisfies large deviation principle (LDP) on C with the rate function 2ϕ(z)
(see [DS89, 1.1.5]). Let us denote
Z
1
n
ctn := ( |z|2nt e−2nϕ (z)dz)− 2
C
for t ∈ [0, 1]. Then by Varadhan’s lemma [DS89, Theorem 2.1.10] and (2.1), for every
t ∈ [0, 1]
1
− lim log cntn = sup(t log r − ϕ(r))
n→∞ n
r>0
= sup(ts − ϕ(es ))
s∈R
=: u(t)
where u(t) (being Legendre-Fenchel transform of ϕ) is a lower-semicontinuous convex
function. Moreover, Legendre-Fenchel transform of u(t) coincides with convex hull of
Φ(s) which is nothing but Φe (s) defined in (2.2). Thus, for every s ∈ R there exists
t0 ∈ [0, 1] such that
st0 − u(t0 ) > Φe (s) − and by lower-semicontinuity there exists an interval J ⊂ [0, 1] containing t0 such that
st − u(t) > Φe (s) − 2.
This implies that for sufficiently large n
1
log(cntn |z|nt ) > ϕe (z) − 3
n
for t ∈ J . Now, letting Jn := {j ∈ {0, . . . , n} : nj ∈ J } we obtain
n
#Jn ≥ |J |
2
for sufficiently large n.
2.1. Bergman kernel asymptotics. Growth condition (2.1) ensures that every polynomial f (z) of degree at most n has finite weighted L2 norm defined by (1.4). We
denote the corresponding Hilbert space of polynomials by (Pn , k · kϕ ). For any given
orthonormal basis {Pjn }nj=0 of Pn Bergman kernel of Pn may be defined as
Kn (z, w) =
n
X
Pjn (z)Pjn (w).
j=0
We also denote the derivatives
n
n
X
X
Kn(1,0) (z, w) =
(Pjn (z))0 Pjn (w) and Kn(1,1) (z, w) =
(Pjn (z))0 (Pjn (w))0 .
j=0
j=0
Let g : C → C be a holomorphic function such that Re(g) = ϕ1 where ϕ1 is the Taylor
polynomial of ϕ of order one. We denote by
u
v
In (u, v) := exp(−n[g(z + √ ) + g(z + √ )]).
n
n
EXPECTED NUMBER OF RANDOM REAL ROOTS
6
We studied Bergman kernel asymptotics in [Bay] (cf. [Chr91, Lin01, Del98, Ber09]).
The following near diagonal asymptotics of Bergman kernel was obtained in [Bay,
Theorem 2.3]:
Theorem 2.2. Let ϕ : C → R be a C 2 weight function satisfying (2.1) and z ∈ Bϕ be
fixed. Then
1
u
v
1
1
(2.4)
Kn (z + √ , z + √ )In (u, v) →
∆ϕ(z) exp( ∆ϕ(z)uv)
n
2π
2
n
n
∞
in C topology on compact subsets of Cu × Cv .
(1,0)
Next, we use Theorem 2.2 in order to obtain diagonal asymptotics of Kn
(1,1)
Kn :
and
Theorem 2.3. Let ϕ : C → R be a C 2 weight function satisfying (2.1) then
1
∂ϕ
(2.5)
n−2 Kn(1,0) (z, z)e−2nϕ(z) → ∆ϕ(z) (z)
π
∂z
and
1
∂ϕ ∂ϕ
(2.6)
n−3 Kn(1,1) (z, z)e−2nϕ(z) → ∆ϕ(z)(2 (z) (z)
π
∂z
∂z
uniformly on compact subsets of Bϕ . In particular, if ϕ is radially symmetric then for every
real interval [a, b] b Bϕ
q
r
(1,1)
(1,0)
Kn (x, x)Kn (x, x) − (Kn (x, x))2
1
1
√
(2.7)
→
∆ϕ(x)
Kn (x, x)
2
n
uniformly on [a, b].
Proof. We fix a compact subset K ⊂ Bϕ and fix z ∈ K. Then by Theorem 2.2 we have
∂ −1
v
1
u
1
[n Kn (z + √ , z + √ )In (u, v)] →
(∆ϕ(z))2 v exp( ∆ϕ(z)uv)
∂u
4π
2
n
n
where the left hand side (LHS) is given by
v
1
u
v
u
1
u
√ Kn(1,0) (z + √ , z0 + √ )In (u, v) − √ Kn (z + √ , z + √ )In (u, v)g 0 (z + √ )
LHS =
n n
n
n
n
n
n
n
In particular,
∂ϕ
1
u
v
1
(2.8)
n−2 Kn(1,0) (z + √ , z + √ )In (u, v) → ∆ϕ(z) (z) exp( ∆ϕ(z)uv)
π
∂z
2
n
n
∂
in C ∞ topology on compact subsets of Cu × Cv . Now, passing to ∂v
on both sides and
using (2.8) together with (2.4) we see that
1 (1,1)
u
v
2
∂ϕ ∂ϕ
1
Kn (z + √ , z + √ )In (u, v) =
∆ϕ(z) (z) (z) exp( ∆ϕ(z)uv)
3
n
π
∂z
∂z
2
n
n
1 1
1
1
(2.9)
+
(∆ϕ(z))2 + ( ∆ϕ(z))3 uv exp( ∆ϕ(z)uv) + o(1)
nπ 4
2
2
as n → ∞.
Finally, using (2.4), (2.8), (2.9) and the assumption ϕ is radially symmetric we
obtain (2.7).
EXPECTED NUMBER OF RANDOM REAL ROOTS
7
2.2. Distribution of complex zeros. In this section we assume that anj are independent copies of the real Gaussian NR (0, 1). We denote the number of complex zeros of a
random polynomial fn in an open set U by NU (fn ). The main result of this section indicates that with high probability almost all zeros of a random polynomial is contained
in the bulk Bϕ . A special case was obtained by Sodin and Tsirelson [ST05] for random
(complex) Gaussian analytic functions (see also [SZZ08] for analogue in the setting
of holomorphic sections). The results of [ST05, SZZ08] are stronger in the sense that
the speed of convergence in n2 . Here, we give a simpler argument at the cost of losing
the optimal estimate in the speed.
Theorem 2.4. Let > 0 and U b C such that ∂U has zero Lebesgue measure. Then
Z
1
(2.10)
P robn fn : | NU (fn ) −
∆ϕe (z)| > ] ≤ exp(−CU, n) for n 1.
n
U
First, we prove a preparative lemma:
Lemma 2.5. Let > 0 and U b C such that ∂U has zero Lebesgue measure. Then for
sufficiently large n there exists En ⊂ Pn such that P rob(En ) ≤ e−C n and
| log |f (z)| − ϕe (z)| ≤ for every f ∈ Pn \ En and z ∈ U
Proof. It follows from [BL15, §6] that
on C. In particular, we have
|
1
2n
log Kn (z, z) converges to ϕe locally uniformly
1
log Kn (z, z) − ϕe (z)| ≤
2n
2
for n 1 and z ∈ U . Now, writing
log |f (z)| = log |ha, u(z)i| +
1
log Kn (z, z)
2n
where u(z) = (uj (z))nj=0 is a unit vector in Cn+1 and a = (aj )nj=0 is the “random
coefficient vector” of f (z). By triangular inequality we have
1
1
| log |f (z)| − ϕe (z)| ≤ | log |ha, u(z)i|| + .
n
n
2
Thus, it is enough to show that
1
P rob[ | log |ha, u(z)i|| > ] ≤ exp(−C n)
n
2
for every z ∈ U . First, we obtain the upper bound:
1
1
P rob[ log |ha, u(z)i| > ] ≤ P rob[(aj )nj=0 : kak > exp( n)]
n
2
2
n
≤ (n + 1)P rob[|aj | > exp( ) for some j]
2
1
≤ (n + 1) exp(− exp(n)).
2
EXPECTED NUMBER OF RANDOM REAL ROOTS
8
On the other hand, since u(z)
P is a unit vector we have the small ball probability of
Gaussian random variable nj=0 aj uj (z)
n
P rob[|ha, u(z)i| < exp(− )] . exp(−n).
2
which finishes the proof.
Proof of Theorem 2.4. Fix > 0 and let χU denote indicator function of U . Then we
can find smooth functions ψ ± : C → R with compact support such that 0 ≤ ψ − ≤ χU ≤
ψ + ≤ 1 satisfying
Z
Z
−
ψ dµe and
ψ + dµe ≤ dµe (U ) + .
µe (U ) − ≤
C
C
+
Then applying Lemma 2.5 on supp(ψ ) we obtain
Z
X
1
1
1
1
+
NU (fn ) ≤
ψ (ζi ) =
log |fn (z)|∆(ψ + (z))
n
n
2π C n
{ζi :fn (ζi )=0}
Z
1
≤
ϕe (z)∆(ψ + (z)) + 2π C
≤ µe (U ) + 2
for f ∈ Pn \ En .
One can obtain the lower estimate by using a similar argument with ψ − . Since µe is
absolutely continuous with respect to Lebesgue measure this completes the proof. 3. P ROOF
OF
T HEOREM 1.1
3.1. Proof of real Gaussian case. It follows from [EK95] (see also [Van15, LPX15])
that for every measurable set E ⊂ R the expected number of real zeros of fn in E is
given by
Z
(3.1)
ENn (E) =
gn (x)dx
E
where
(3.2)
gn (x) =
q
(1,1)
(0,1)
1 Kn (x, x)Kn (x, x) − (Kn (x, x))2
π
Kn (x, x)
and
n
n
n
X
X
X
n 2 2j
(0,1)
n 2 2j−1
(1,1)
Kn (x, x) =
(cj ) x , K
(x, x) =
j(cj ) x
, K
(x, x) =
j 2 (cnj )2 x2j−2
j=0
j=0
j=0
for x ∈ R. Moreover, by Theorem 2.3 for every real interval [a, b] b Bϕ
r
1
1√
gn (x) =
n
∆ϕ(x) + o(1) as n → ∞
π
2
uniformly on [a, b].
Hence, the assertion follows from additivity of ENn on R, absolute continuity with
respect to Lebesgue measure and Theorem 2.4.
EXPECTED NUMBER OF RANDOM REAL ROOTS
9
3.2. Non-Gaussian case. In what follows we let Dr (z) denote a disc in complex plane
of radius r and center z. For a complex random variable η we denote its concentration
function by
Q(η, r) := sup P rob{η ∈ Dr (z)}.
z∈C
We say that η is non-degenerate if Q(η, r) < 1 for some r > 0. If η and ξ are independent
complex random variables and r, c > 0 then we have
r
(3.3)
Q(η + ξ, r) ≤ Q(η, r) and Q(cζ, r) = Q(ζ, )
a
In order to prove the statement for non-Gaussian ensembles we use the replacement
principle of [TV15, Theorem 3.1]. We verify the hypotheses of [TV15, Theorem 3.1]
by proving sufficient conditions introduced in [TV15, §4].
Concentration of log-magnitude. First, we prove the following result which verifies
[TV15, Theorem 3.1(i)].
Proposition 3.1. For fixed > 0, z0 ∈ C and r > 0
1
1
P rob[| log |fnζ (z)| − ϕe (z)| > ] = O( √ )
n
n
for every z ∈ Dr (z0 ) \ {0} and sufficiently large n.
Proof. The upper bound follows from the argument given in Lemma 2.5 where we
replace the Gaussian by ζ and use the moment condition E|ζ|2+δ < ∞.
Now, we prove the lower bound by using an idea from [KZ]. By Proposition 2.1 that
for every > 0 there exists an interval J ⊂ [0, 1] such that
cnj |z|j > en(ϕe −)
for j ∈ Jn := {1 ≤ i ≤ n :
i
n
∈ J }. Next, we define
X
X
Xn =
αjn anj and Yn =
αjn anj
j∈Jn
j6∈Jn
where
αjn = e−n(ϕe −) cnj |z|j .
Then by (3.3) and sufficiently large n we have
P rob[|fn (z)| < en(ϕe (z)−2) ] ≤ Q(Xn + Yn , e−n ) ≤ Q(Xn , e−n ).
Now, it follows from Kolmogorov-Rogozin inequality [Ess68] and αjn > 1 for j ∈ Jn
that
X
1
1
1
Q(Xn , e−n ) ≤ C(
(1 − Q(αjn anj , e−n ))− 2 ≤ C1 |Jn |− 2 ≤ C2 n− 2 .
j∈Jn
Next, combining with the argument given in the proof of Theorem 2.4, it follows
from Proposition 3.1 that
1
1
P rob[| NDr (z0 ) − µe (Dr (z0 ))| > ] = O( √ ).
n
n
EXPECTED NUMBER OF RANDOM REAL ROOTS
10
This together with [TV15, Proposition 4.1] gives [TV15, Theorem 3.1(ii)].
Comparibility of log-magnitude. An immediate consequence of Proposition 2.1 and
[TV15, Theorem 4.5] gives [TV15, Theorem 3.1(iii)]
Repulsion of zeros. It remains to prove weak repulsion of zeros for ζ = NR (0, 1). For
(k,l)
any natural numbers k, l we denote the the mixed (k, l)-correlation function ρf :=
Rk × (C \ R)l → R+ which is defined by
X X
ψ(ζi1 , . . . , ζik , ηj1 , . . . , ηjl )
E
i1 ,...,ik j1 ,...,jl
Z
Z
(k,l)
=
Rk
(C\R)l
ψ(x1 , . . . , xk , z1 , . . . , zl )ρf
(x1 , . . . , xk , z1 , . . . , zl )dx1 . . . dxk dz1 . . . dzl
where ζ1 , . . . ζk denote the real roots and η1 , . . . ηk denote the complex roots of Gaussian
random polynomial fn .
It follows from [TV15, Lemma 4.6] that it is enough to prove that for x ∈ Bϕ ∩ R
and sufficiently small > 0 the real repulsion estimate
(2,0)
(3.4)
ρf˜ (x, x + ) = O()
and the complex repulsion estimate
(0,1)
(3.5)
ρf˜
= (x +
√
−1) = O()
hold for the scaled random polynomials f˜n (z) := fn ( √zn ). As the estimates (3.4) and
(3.5) are standard in the literature and follow from Kac-Rice formula and Bergman
kernel asymptotics in §2 (see e.g. the argument given in [TV15, §11], see also [BD04,
§7]) we omit the details.
4.
EXAMPLES
In this section we provide an application of Theorem 1.1. We consider the circular
weight functions of the form
ϕ(z) = −α log |z| + |z|β
where α ≥ 0 and β > 0. Then it follows from [ST97, pp. 245]) that


|z| ≤ r0
ϕ(r0 )
ϕe (z) = ϕ(z)
r0 < |z| < R0

log |z| + ϕ(R ) − log R |z| ≥ R
0
0
0
1
1
where r0 = ( αβ ) β and R0 = ( 1+α
) β . In particular, the weighted equilibrium measure
β
dµe =
1Dϕ
2π
β 2 rβ−1 drdθ, z = reiθ
EXPECTED NUMBER OF RANDOM REAL ROOTS
11
is supported on the annulus Dϕ = {r0 ≤ |z| ≤ R0 }. In this setting a random polynomial
is of the form
n
X
fn (z) =
anj cnj z j
j=0
where
cnj
Z
1
β
= ( |z|2j+2nα e−2n|z| dz)− 2
C
and anj are independent copies of a random variable ζ with mean zero, variance one
and satisfying E|ζ|2+δ < ∞ for some δ > 0.
Finally, it follows from Theorem 1.1 that asymptotic number of real roots is given
by
√
√
1
2β √
lim √ ENnζ =
( α + 1 − α).
n→∞
π
n
R EFERENCES
[Bay]
[BD04]
[Ber09]
[BL15]
[BP32]
[Chr91]
[Das71]
[Del98]
[DS89]
[EK95]
[EO56]
[Ess68]
[ET50]
[IM71a]
[IM71b]
[Kac43]
[KZ]
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