resume03_2. 68KB Jun 04 2011 12:07:02 AM

Journal de Th´

eorie des Nombres

de Bordeaux

20

_{(2008), 281–287}

Extensions of the Bloch–P´

olya theorem on the

number of real zeros of polynomials

par

Tam´

as ERD ´

ELYI

R´esum´e. _{Nous prouvons qu’il existe des constantes absolues}c1>

0 etc2>0 telles que pour tout

{a0, a1, . . . , an} ⊂[1, M], 1≤M ≤exp(c1n1/4),

il existe

b0, b1, . . . , bn ∈ {−1,0,1}

tels que

P(z) =

n X

j=0 bjajzj

a au moinsc2n1/4

changements de signe distincts dans ]0,1[. Cela am´eliore et ´etend des r´esultats ant´erieurs de Bloch et P´olya.

Abstract. _{We prove that there are absolute constants c1 > 0}

andc2>0 such that for every

{a0, a1, . . . , an} ⊂[1, M], 1≤M ≤exp(c1n 1_/4

),

there are

b0, b1, . . . , bn ∈ {−1,0,1}

such that

P(z) =

n X

j=0 bjajzj

has at leastc2n1/4

distinct sign changes in (0,1). This improves and extends earlier results of Bloch and P´olya.

Tam´asErd´elyi

Department of Mathematics Texas A&M University College Station, Texas 77843

E-mail:terdelyi@math.tamu.edu

URL:http://www.math.tamu.edu/∼tamas.erdelyi