The Budan-Fourier Theorem

m

2

3

4

5

new polynomial

a

-5

b

5

zoom

p(x)

2

x

+2x-2

p'(x)2x+2

Budan–Fourier sequence



2

x

+2x-2,2x+2,2

	p 0	p 1	p 2
a = -5	13.00	-8.00	2.000
b = 5	33.00	12.00	2.000

signs

	p 0	p 1	p 2
a	+	-	+
b	+	+	+

V(-5)2, V(5)0, V(a) - V(b) = 2

Given a polynomial

p(x)

of degree

n

, the sequence

p

0

(x)=p(x)

,

p

1

(x)=



p

(x)

, ...,

p

n

(x)=

(n)

p

(x)

is called the Budan–Fourier sequence of

p(x)

.

Let

N

be the number of real roots of

p(x)=0

over an open interval

(a,b)

(i.e. excluding

a

and

b

). Then

N=V(a)-V(b)-ν

, where

V(a)-V(b)

is the difference between the number of sign changes of the Budan–Fourier sequence evaluated at

a

and at

b

, and

ν

is a non-negative even integer. Thus the Budan–Fourier theorem states that the number of roots in the interval is equal to

V(a)-V(b)

or is smaller by an even number.