Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
Tamás Erdélyi1
1 Department of Mathematics Texas A&M University College Station, Texas 77843
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 281-287.

We prove that there are absolute constants c 1 >0 and c 2 >0 such that for every

{a0,a1,...,an}⊂[1,M],1≤M≤exp(c1n1/4),

there are

b0,b1,...,bn∈{-1,0,1}

such that

P(z)=∑j=0nbjajzj

has at least c 2 n 1/4 distinct sign changes in (0,1). This improves and extends earlier results of Bloch and Pólya.

Nous prouvons qu’il existe des constantes absolues c 1 >0 et c 2 >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)=∑j=0nbjajzj

a au moins c 2 n 1/4 changements de signe distincts dans ]0,1[. Cela améliore et étend des résultats antérieurs de Bloch et Pólya.

Received:
Published online:
DOI: 10.5802/jtnb.627
Tamás Erdélyi 1

1 Department of Mathematics Texas A&M University College Station, Texas 77843 
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Tamás Erdélyi. Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 281-287. doi : 10.5802/jtnb.627. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.627/

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