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, p. 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],1Mexp(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],1Mexp(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.

@article{JTNB_2008__20_2_281_0,
     author = {Erd\'elyi, Tam\'as},
     title = {Extensions of the Bloch--P\'olya theorem on the number of real zeros of polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     pages = {281-287},
     doi = {10.5802/jtnb.627},
     mrnumber = {2477504},
     zbl = {1163.11022},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_2_281_0}
}
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://www.numdam.org/item/JTNB_2008__20_2_281_0/

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