Zeros of random Müntz polynomials

We study the expected number of positive zeros of Müntz polynomials with real i.i.d. coefficients. For the standard Gaussian coefficients, we establish asymptotic results for the expected number of positive zeros when the exponents of Müntz monomials that span our random Müntz polynomials have polyn...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 552; no. 2; p. 129799
Main Authors Lubinsky, Doron S., Pritsker, Igor E.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.12.2025
Subjects
Expected number of real zeros
Fewnomials
Lacunary polynomials
Müntz polynomials
Random coefficients
Sparse polynomials
Müntz polynomials
Lacunary polynomials
Fewnomials
Sparse polynomials
Expected number of real zeros
Random coefficients
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ISSN0022-247X
DOI10.1016/j.jmaa.2025.129799

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Summary:We study the expected number of positive zeros of Müntz polynomials with real i.i.d. coefficients. For the standard Gaussian coefficients, we establish asymptotic results for the expected number of positive zeros when the exponents of Müntz monomials that span our random Müntz polynomials have polynomial and logarithmic growth. We also present many bounds on the expected number of zeros of random Müntz polynomials with various real i.i.d. coefficients, including the case of arbitrary nontrivial real i.i.d. coefficients. Since Müntz polynomials include lacunary polynomials, sparse polynomials or fewnomials as special cases, our results directly apply to the expected number of real zeros for those classes of polynomials with gaps.
ISSN:0022-247X
DOI:10.1016/j.jmaa.2025.129799