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Pseudo-boolean functions and the multiplicity of the zeros of polynomials

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Journal d'Analyse Mathématique Aims and scope

Abstract

A highlight of this paper states that there is an absolute constant c 1 > 0 such that every polynomial P of the form P(z) = Σ n j=0 a j z j, a j ∈ ℂ with

$$\left| {a_0 } \right| = 1, \left| {a_j } \right| \leqslant M^{ - 1} \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right), j = 1,2, \ldots ,n,$$

for some 2 ≤ M ≤ e n has at most \(n - \left\lfloor {{c_1}\sqrt {n\log M} } \right\rfloor \) zeros at 1. This is compared with some earlier similar results reviewed in the introduction and closely related to some interesting Diophantine problems. Our most important tool is an essentially sharp result due to Coppersmith and Rivlin asserting that if F n = {1, 2, …, n}, there exists an absolute constant c > 0 such that

$$\left| {P(0)} \right| \leqslant \exp (cL)\mathop {\max }\limits_{x \in {F_n}} \left| {P(x)} \right|$$

for every polynomial P of degree at most \(m \leqslant \sqrt {nL/16} \) with 1 ≤ L < 16n. A new proof of this inequality is included in our discussion.

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  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Tamás Erdélyi

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  1. Tamás Erdélyi
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Erdélyi, T. Pseudo-boolean functions and the multiplicity of the zeros of polynomials. JAMA 127, 91–108 (2015). https://doi.org/10.1007/s11854-015-0025-1

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  • DOI: https://doi.org/10.1007/s11854-015-0025-1

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