Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

Acta Arithmetica, Tome 172 (2016), p. 271-284 / Harvested from The Polish Digital Mathematics Library

Résumé

For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_{p} (n, L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L (\sum_{j = 1}^{n} | a_{j} {|^{p})}^{1 / p}$ , $a_{j} \in ℂ$ , such that ${(x - 1)}^{k}$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let $μ_{q} (n, L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $| Q (0) | > 1 / L (\sum_{j = 1}^{n} {| Q (j) |}^{q})^{1 / q}$ . We find the size of $κ_{p} (n, L)$ and $μ_{q} (n, L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about $μ_{\infty} (n, L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.

Publié le : 2016-01-01

Zbl 06545352

EUDML-ID : urn:eudml:doc:279368

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     title = {Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1},
     journal = {Acta Arithmetica},
     volume = {172},
     year = {2016},
     pages = {271-284},
     zbl = {06545352},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8129-11-2015}
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 (éd.). Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1. Acta Arithmetica, Tome 172 (2016) pp. 271-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8129-11-2015/