Download The Number of Real Roots of Random Polynomials of Small Degree

Indian Statistical Institute
The Number of Real Roots of Random Polynomials of Small Degree
Author(s): William B. Fairley
Source: Sankhyā: The Indian Journal of Statistics, Series B, Vol. 38, No. 2 (May, 1976), pp. 144
-152
Published by: Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25052004
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Indian Journal of Statistics, Series B.
http://www.jstor.org
: The
Sankhy?
1976, Volume
Journal
Indian
of Statistics.
144-152.
Series B, Pt.
2, pp.
38,
THE NUMBEROF REAL ROOTS OF RANDOM
POLYNOMIALS OF SMALL DEGREE*
B. PAIRLEY
By WILLIAM
School
Kennedy
SUMMARY.
number
Random
of authors,
of Government,
random
with
polynomials
Kac
Harvard
1949), who
University
coefficients
that
showed
have
the average
been
studied
number
of real
by a
roots
(1943,
including
of polynomials
of degree n is asymptotically
the aver
The present
paper
(2/tc) log n.
investigates
? 1with
of real roots for polynomials
of small degree and coefficients
that are 1 or
age number
equal
of the average
exact distributions
behavior
for small n is showTi by finding
probability.
Log-like
1 and 10 and by sampling
of the numbers
of real roots for n between
the large but finite populations
for n between
10 and
50.
1.
Introduction
are given accord
is a polynomial
whose
coefficients
polynomial
a
to
some
Denote
distribution.
ing
polynomial
by p(x), and let
probability
:
the coefficients
be denoted
a%, so that for an w-th degree polynomial
A random
p(x)
The
has
distribution
been
=
of the number
of real
N
several
studied
...
a0+a1x+a?xP-\-...+ant??.
roots
of random
and
in particular
several different
the mean
(1)
polynomials
of
number
authors,
by
has been studied
for
of the
distributions
En(N),
and
is
An
reference
coefficients
who
that
Bloch
showed
a\.
(1932)
early
Polya
=
=
=
=
=
when
-1]
0]
P(at
P[at
P[a{
En(N)
1]
O(V^)
1/3. Little
real
roots,
wood and Offord (1939) gave the upper bound En(N) <
for three
[?1,
Then
=
distributions
1]; and a% normally
Kac
(1943, 1949)
?
: (i) P[a$
distributed.
showed
that
?
=
25 (logn)2+12
log n
=
on
P[?|
1]
1/2, (ii) a% uniform
The a? are independent
in each case.
for uniform
and normally
distributed
1]
as the degree
the asymptotic
value for En(N)
coefficients
to infinity
is given by the simple
increases
n of the polynomial
expression
En(N)^(2l7T)logn.
...
(2)
are
few real roots of random
and
Erdos
remarkably
polynomials.
?
that
this
case
formula
holds
for
the
Offord (1956) showed
asymptotic
P[ai
1]
= ?
=
=
and
Stevens
and
Maslova
(1965)
P[a{
1]
Ibragimov
1/2.
(1971)
classes of distributions
this result to wider
have extended
within
the domain
There
of attraction
*This
to Harvard
of the normal.
research
University.
was
facilitated
by
a grant
from
the National
Science
Foundation,
GS-2044X,
145
RANDOM POLYNOMIALS OF SMALL DEGREE
2.
This
nomials
babilities
populations
Finite
on an
paper reports
of small degree
and mutually
and
We
result
the asymptotic
above,
are particularly
to study
amenable
and
(2) holds for ?1
polynomials,
for small values of n.
mentioned
they
of En(N)
for poly
that are 1 or ?1 with
equal pro
As
call these ?1
polynomials.
of the values
investigation
for coefficients
independent.
polynomials
of
there
of all -j-1 polynomials
of given degree n is finite,
population
of
2W+1 polynomials
the
to
2n+1
the
choices
of
sequences
corresponding
?
1 or
1. For very small values of n the entire exact distri
coefficients
The
being
n-\-\
of the number
butions
with
these,
calculations
cients
all
-1/2]
and En(N)
may be obtained,
the real roots of all the 2W+1polynomials.
roots N
simply by calculating
are reduced by certain
and roots,
of
of real
the
real
symmetries
5 below
in Section
out
spelled
roots of
these polynomials
in the relations
along
The
between
coeffi
the fact
and
simplified by
the real intervals
lie within
that
[?2,
U [1/2, 2].
Of course
as n increases
comes
there
a point where calculating
the roots of
for
To obtain the exact distribution
all 2W+1 polynomials
too costly.
becomes
=
of degree
n
the roots of 213 =8192
15 would require calculating
polynomials
of degree
20. The present
15 and for n = 20, 218 = 260, 544 polynomials
of poly
sample survey design to sample the populations
study uses a stratified
as n increases.
for n larger than 10 in order to study En(N)
nomials
3.
Exact
distributions
n=
of
distributions
n =
1, 2, ..., 10.
1 plots the exact frequency
1, 2, ..., 10, figure
and degrees
of real roots for ?1
the numbers
polynomials
1 gives the computed
values.
Table
For
TABLE
1. EXACT DISTRIBUTIONS
FOR
DEGREES
OF NUMBERS
n =
1, 2,
OF REAL ROOTS
..., 10
odd degree
degree
n
1
number
real roots
of
count
4
1
degree
frequency
1.000
n
even
number
of
real roots
degree
count
40
2
4
3
12 1
.750
4
.250
34
5
7
44 1
1
164
012
2
.688
.312
3 20
.641
.359
392
6
.582
.414
10
.625
5961
424
3
.004 54
.500
2
.600
.375
20
032
.250
0 116
.742
.227
96
.7502
8
2 380
4
16
9
frequency
2
0408
.738
1512
4
.063
128
.031
.199
146
WILLIAM B. FAIBLE Y
odd
I 3
1 3
n=
*^
?
1 3
7
??s
)
number
of roots
1 3 5
v =9
even degree
?
0
0
2
0
J_
2~ 4
~?
2
n=
1.
Fig.
Exact
n=
Solutions
for
the
Distributions
roots
descent
to locate
steepest
find closer approximations.
by F. Lilley
of real roots
separately,
...,
1, 2,
of Numbers
? =
8
of Real
number
of roots
I
2 4
?
Roots
'O
For Degrees
10.
were
obtained
approximate
The method
that employs
using an algorithm
to
roots and then Newton's
method
and
computer program
of
distributions
are described
the numbers
The means
of the exact
(1967).
n
are plotted
even and n odd
2. Treating
in Figure
the cases
the plot indicates
in the means with n.
smooth
increases
Bn(N)
u
10
degree
Fig.
2.
Means
of Exaet
Distributions
of Numbers
of R^al
Roots,
RANDOM POLYNOMIALS OF SMALL DEGREE
4.
experiment
Sampling
for
147
n
larger
10 a sampling
of
takes advantage
experiment
a
n
n
n
even.
of
values
of
this smoothness
for
odd and
grid
by taking only
of polynomials
the population
for each degree n was sampled.
To estimate En(N)
was chosen to esti
random
allocation
A stratified
sample using proportional
For
of n larger
values
than
rule
A stratifying
variable
En(N)
efficiently.
an
states that
of signs, which
upper bound
real
roots
mate
is suggested
by
for the number
Descartes'
of positive
of changes of sign in the sequence of
of a polynomial
is the number
are 1 or ?1,
to the absolute
the coefficients
coefficients.
When
\p(l) | is equal
of +l's
of ? l's
between
the number
and the number
value of the difference
it
of sign changes and therefore
in turn is associated
with
the number
of positive
the number
real roots.
The
is
with
is conjectured,
conjecture
?
n
a
for degrees
verified
N and
Since
1, 2, ..., 10 by computing
|^(l)j.
a
root
of
is
root
of
is
associated
1) J negatively
p(x)
negative
p(?x),
p(?
positive
\
? 1 are
1 ) |and p(
real roots. Now
with the number of negative
))
p(
independent
|
|
which
for n odd, and for n even they are less and less correlated as n increases, so that
=
a?
the sum ^ST
that is negatively
associated
Jjp(l) +|
jis random variable
|i>(?1)
or negative.
of real roots positive
with the number
Therefore K is suggested
as a simply computed
variable
for sampling ?1
stratifying
polynomials.
?
100 polynomials
sample of m
of K for each of 10 populations
n =
of n between
15 and n =
A
values
value
between
n =
16 and n =
of higher degrees
ing roots mount
than
these
was taken from the strata defined by
of polynomials,
for every eighth
namely
?
47, that is, n
15, 23, 31, 39, 47, and
?
n
48.
For polynomials
16, 24, 32, 40,
is,
48, that
the costs
and the numerical
difficulties
of comput
sharply.
in Section
5 on the partitioning
of the popula
the results presented
Using
4
8
same
the
of roots,
and of
tions into groups of
numbers
having
polynomials
of each degree.
to sample a quarter of the 2W+1polynomials
it was only necessary
in Table 2 below refer only to
(For this reason too, the values of K appearing
sums
of the
last n?1
coefficients
of p(x).
See Section
5.)
2 gives the observed mean numbers,
ynjt, of real roots for the different
strata.
The negative
association
between
of yn^ and of K for a
the values
n
were
is
The
values
of
estimated
from the stratum
apparent.
given
EJN)
:
means
means,
yn
by the weighted
Table
yn
where
M
is the population
size
=
(M
XMKynKIM
K
=
(2^+1/4)
=
2?_1).
148
WILLIAM B. FAIRLEY
TABLE
2.
STRATUM MEANS, ynK
n
degree
K
>
15
23
31
39
47
2
2.47
2.00
2.47
3.33
3.40
6
1.81
2.20
2.45
2.64
3.40
6
1.61
10
2.43
2.10
3.00
3.00
>10
2.00
2.13
2.55
2.50
14
2.43
>14
18
2.20
1.25
>18
degree
K
n
16
24
32
40
48
2
2.13
2.00
3.00
2.75
2.40
4
2.00
2.44
3.00
2.44
2.57
6
2.17
2.50
2.67
3.29
2.33
8
1.79
2.00
2.35
2.88
2.89
10
2.20
2.00
2.71
2.43
2.50
2.00
1.67
1.00
2.57
14
2.22
2.89
2.20
>14
1.45
2.50
1.50
2.20
2.00
1.50
>10
12
1.20
>12
16
>16
The
values
the weighted
in Table
3. The
Sn, are given
used to compute
of
the variances
yn, and their estimated
in each stratum,
sample variances
:
to
the
formula
s% according
means,
1 VMR{MK-mk)
si= M
^L_
K
mu
variances,
s%K,were
149
RANDOM POLYNOMIALS OF SMALL DEGREE
TABLE
3. ESTIMATES OF MEANS AND VARIANCES
STRATIFIED SAMPLES
even
odd degree
mean
degree
n
yn
mean
$2
variance
FROM
degree
vaiance
yn
15
1.98
.88
16
2.00
1.36
23
2.22
1.43
24
2.10
1.15
31
2.28
1.32
32
2.40
1.44
39
2.78
1.99
40
2.50
1.39
47
2.86
1.86
48
2.40
1.71
?2
in Figure
3 for n even, n =
values,
16, 24,
yn, are plotted
32, 40, 48. The exact values of En(N) for the smaller even degree polynomials,
^ = 2,4,6,8,10
are also plotted.
values
The
asymptotic
(2/7r) log n of
in Figure
is also given
3. The exact and estimated
of En(N)
values
En(N)
The
estimated
a
as n
behavior
log-like
the behavior
is similar.
reveal
n odd
?
even
increases
for
small
values
of n.
For
logn
Legend
exact
x estimated
-L
3.
Fig.
5.
If p(x)
then
p(x)
in reverse
three
Numbers
Mean
Partitioning
has N
roots
X
30
of Roots
the
then ? p(x)
40
Exact
and
Estimated
populations
of
and p(?x)
have N
Ifx is a root of the polynomial
p(x) whose
_L
60
50
Degree
n even
polynomials
roots.
coefficients
If # is a root of
are those of p(x)
so that p(x) also has N roots.
of these
combinations
Various
of a given polynomial
transformations
p(x) lead to still other polynomials
B2-7
order,
150
WILLIAM B. FAIRL?Y
having N roots
the polynomials
: ? p(~x),
and ? p(?x).
?p(x),
p(?x),
have coefficients
of l's and ?l's.
of
products
in the population.
these
Further
nomials
in algebraic
do
transformations
In our application
not
lead
to other
poly
be
described
may
conveniently
of transformations
acting on the set S of all
In fact the situation
as a group
of l's and ?l's
of length
terms
The group T is a set of eight trans
n+l.
on the set S, ti :S-?S,
or mappings,
where
?0, tx, ...,t7, defined
the
is composition
of mappings,
the group operation
and, using
polynomial
l's and ? l's, the ti are defined by
notation p(x) to denote a sequence of n+1
sequences
formations
n even n odd
tQp(x)
=
=
hp(x)
t2p(x)
=
=
tsp(x)
=
hP(x)
t5p(x)
t6p(x)
=
=
p(x)
?
p(x)
identity
same
changes
sign of all coefficients
p(?x)
changes
sign of all odd coefficients
p(x)
reverses
order of coefficients
?p(?x)
changes
of sign of even
?p(x)
reverses
order and changes
reverses
order and
p(?x)
same
same
?
reverses
~P(~~~X)
order
changes
element
for all
cation
of this
same
and
(odd)
group is idempotent,
is its own inverse.
element
is; for n even
ta
h
h
h
h
h
h
U
same
changes
8-element
i, so that each
table for the group
U
tes
i.e.,
?c
fe
?3
?4
tcz
tR
h
T have
in the orbit
the
same
t0, the
complete
identity,
multipli
h
In
*f\
h
te
symmetric
commutativity
of this
=
q(x)
tip(x)
a common orbit under
the set 8, is divided by
of 2n+1 polynomials,
A population
or
orbits such that if p(x) e S is in an orbit,
group into subsets
is also
=
:
t?
tn
?^
The
ta
Pc
same
all signs
(even)
(even) signs
Each
same
coefficients
(odd) signs
t7p(x)
same
for all t^T.
All polynomials
roots.
of
real
number
within
the action
then
151
RANDOM POLYNOMIALS OF SMALL DEGREE
n odd
For
and n >
are the
2 there
cases
following
:
?
=
then p(x) = p(x), i.e., t3
t0 so that also t5
(i) p(x) symmetric,
==
of p(x)
consists
of only 4 polynomials,
t7
?4 and the orbit
?
and
,
p(?x).
?p(x),
p(x)
p(?x),
(ii)
w/2
odd
even,
of p(x)
coefficients
are
then
anti-symmetric,
of 4 polynomials.
consist
p(x)
?
=
while
symmetric
=
i.e., ?3
?4 and
p(?x),
tl916
namely
even
the
=
t2,
:
coefficients
orbit
of p(x)
while
of p(x) symmetric
odd coefficients
(iii) n/2 odd, even coefficients
=
=
are anti-symmetric,
then p(x)
i.e., ?3
t2 and the orbit of p(x) con
p(?x),
sists of 4 polynomials.
(iv)
p(x)
in all three
asymmetric
senses
above,
then
is in an orbit
p(x)
of 8 polynomials.
a population
consists of orbits of 8 or 4 whose
of 2^+i polynomials
of real roots.
from
have the same numbers
Simple random sampling
can be performed
more efficiently
the population
sampling
by simple random
Thus
members
of entering
orbits of 8 are given twice the probability
provided
is confined
the sample as are orbits of 4. This will be the case if sampling
are
to a list of polynomials
first
two
fixed at one
coefficients
whose
identical,
of the orbits,
of (1, 1), (?1, ?1),
It may be verified
?1,
(
that
are two
representatives
random
Stratified
4.
list of
for concreteness,
1), (1, ?1).
(1, 1) is chosen.
Suppose
=
there
list of 2n~1
in the resulting
2w+1/4 polynomials
of each orbit of size 8 and one of each orbit of size
sampling
1/4 of the population
be carried
would
out on a stratification
of this
of polynomials.
of the above theory, for ft = 5 the set S consists of 2n+1
As an illustration
=
64 polynomials
6 orbits of size 8 and 4 orbits of size 4. The
26
with
?
following
numbers
are the
In parentheses
of the 64.
sign sequences
solution of the
of each orbit as determined
by numerical
is a list of the
of roots
polynomials.
(1) (1)
++++-+
+
_
+++-++
+++-+
+-+-
(3)(3)
(1)
+ + +
+ +-+-
+-+
-+-+++
-+-++
~+-+
-+++-
+-++++
+-+++
++-+++
-++-++
+++-+-
+ +-+-
-1_?h-+
+ -+
+-+++ +
++++
++-++-
-+++H?+
+H-+
+
+-
+-+
-+_++_
-++++
+++-+
+-+
-
+
152 WILLIAM B. FAIRLEY
(1)(1)
+ + + + +
+
+ +
+
++ + + + +
+- +
+
_
+
+-.-_
+
+
-
-+
(1)
(1)(3)
+ + +-+
+- + + -+
_
+ _+
_
+-+
-
+
+-+
_
+-+
+-+
+-h
_
+ + + +
_
+
+
-
+ +-+
-
+
Acknowledgement
The
author
thanks
Frederick
for helpful
Mosteller
comments.
References
A.
Bloch,
and
Math.
P.
Erdos,
and
Proc.
II,
: On
(1943)
American
London
F.
Lilley,
Stevens,
J.
D.
sertation,
Paper
(1971)
with
the average
of
certain
algebraic
London
Proc.
equations.
the number
of real
roots
of a random
algebraic
equation.
: On
the
of
number
expected
zero means.
Theory
zeros
real
and
of Probability
of
random
its Applications,
and
(1965)
New
number
A.
Offord,
Proc
(1939)
Cambridge
: The
average
York
University,
: February,
roots
of a random
algebraic
Bulletin
equation.
of real
roots
of a random
algebraic
(II).
equation
Proc.
390-408.
of a Polynomial.
: Zeros
II.
received
50,
of real
number
314-320.
49,
the average
Soc,
(1967)
equation
N.
Soc,
Math.
Littlewood,
roots
139-160.
6,
Coefficients
Math.
: On
-(1949)
the
2.
No.
XVI,
: On
(1956)
Soc,
I. and Maslova,
polynomials
M.
A.
Math.
I.
: On
(1932)
102-114.
33,
Offord,
London
Ibragimov,
Kac,
G.
Polya,
Soc,
General
: On
the
number
Soc,
35,
133-148.
real
zeros
Phil.
number
1975.
electric
of
Department
information
of real
of a random
of Mathematics.
series,
roots
No.
65SD351.
of a random
polynomial,
algebraic
doctoral
dis