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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| Published in | Journal of mathematical analysis and applications Vol. 552; no. 2; p. 129799 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
15.12.2025
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| ISSN | 0022-247X |
| DOI | 10.1016/j.jmaa.2025.129799 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 129799 |
| Author | Pritsker, Igor E. Lubinsky, Doron S. |
| Author_xml | – sequence: 1 givenname: Doron S. surname: Lubinsky fullname: Lubinsky, Doron S. email: lubinsky@math.gatech.edu organization: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA – sequence: 2 givenname: Igor E. orcidid: 0000-0002-3102-5003 surname: Pritsker fullname: Pritsker, Igor E. email: igor@math.okstate.edu organization: Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA |
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| Cites_doi | 10.1214/17-AOP1219 10.1090/proc/12836 10.1142/S0219199715500522 10.1112/plms/s2-33.1.102 10.1007/BF01456964 10.1090/S0002-9904-1943-07912-8 10.1137/18M1228682 10.1016/j.aim.2006.03.008 10.2140/pjm.1953.3.673 10.1353/ajm.2022.0000 10.1137/1116023 10.1093/imrn/rnu084 10.1090/S0273-0979-1995-00571-9 10.1112/plms/s2-50.5.390 10.1007/s10958-018-3705-4 |
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| Keywords | Müntz polynomials Lacunary polynomials Fewnomials Sparse polynomials Expected number of real zeros Random coefficients |
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| SubjectTerms | Expected number of real zeros Fewnomials Lacunary polynomials Müntz polynomials Random coefficients Sparse polynomials |
| Title | Zeros of random Müntz polynomials |
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