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.

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

{a 0 ,a 1 ,...,a n }[1,M],1Mexp(c 1 n 1/4 ),

there are

b 0 ,b 1 ,...,b n {-1,0,1}

such that

P(z)= j=0 n b j a j z j

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

{a 0 ,a 1 ,...,a n }[1,M],1Mexp(c 1 n 1/4 ),

il existe

b 0 ,b 1 ,...,b n {-1,0,1}

tels que

P(z)= j=0 n b j a j z j

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.

DOI: 10.5802/jtnb.627
Erdélyi, Tamás 1

1 Department of Mathematics Texas A&M University College Station, Texas 77843
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Erdélyi, Tamás. 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. http://archive.numdam.org/articles/10.5802/jtnb.627/

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