Advanced book on Mathematics Olympiad

2.2 Polynomials 59

Chebyshev’s theorem.For fixedn≥ 1 , the polynomial 2 −n+^1 Tn(x)is the unique monic

nth-degree polynomial satisfying

max

− 1 ≤x≤ 1

| 2 −n+^1 T(x)|≤max

− 1 ≤x≤ 1

|P(x)|,

for any other monicnth-degree polynomialP(x).

One says that among all monicnth-degree polynomials, 2−n+^1 Tn(x)has the smallest

variation away from zero on[− 1 , 1 ]. This variation is 2 n^1 − 1. Let us see how Chebyshev’s

theorem applies to a problem fromChallenging Mathematical Problems with Elementary

Solutionsby A.M. Yaglom and I.M. Yaglom.

Example.LetA 1 ,A 2 ,...,Anbe points in the plane. Prove that on any segment of length

lthere is a pointMsuch that

MA 1 ·MA 2 ···MAn≥ 2

(

l

4

)n

.

Solution.Rescaling, we can assume thatl=2. Associate complex coordinates to points

in such a way that the segment coincides with the interval[− 1 , 1 ]. Then

MA 1 ·MA 2 ···MAn=|z−z 1 |·|z−z 2 |···|z−zn|=|P(z)|,

whereP(z)is a monic polynomial with complex coefficients, andz∈[− 1 , 1 ]. Write

P(z)=R(z)+iQ(z), whereR(z)is the real part andQ(z)is the imaginary part of the

polynomial. Sincezis real, we have|P(z)|≥|R(z)|. The polynomialR(z)is monic,

so on the interval[− 1 , 1 ]it varies away from zero at least as much as the Chebyshev

polynomial. Thus we can findzin this interval such that|R(z)|≥ 2 n^1 − 1. This implies

|P(z)|≥ 2 · 21 n, and rescaling back we deduce the existence in the general case of a point

Msatisfying the inequality from the statement. 

Stepping aside from the classical picture, let us also consider the families of polyno-

mialsTn(x)andUn(x)defined byT 0 (x)=2,T 1 (x)=x,Tn+ 1 (x)=xTn(x)−Tn− 1 (x),

andU 0 (x)=1,U 1 (x)=x,Un+ 1 (x) =xUn(x)−Un− 1 (x). These polynomials are

determined by the equalities

Tn

(

z+

1

z

)

=zn+

1

zn

and Un

(

z+

1

z

)

=

(

zn+^1 −

1

zn+^1

)/(

z−

1

z

)

.

Also,Tn(x)=^12 Tn( 2 x)andUn(x)=Un( 2 x). Here is a quickie that usesTn(x).

Example.Letabe a real number such thata+a−^1 is an integer. Prove that for any

n≥1, the numberan+a−nis an integer.