On a class of lacunary almost Newman polynomials modulo p and density theorems

Abstract

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζ_β (z) of the β-shift, for β > 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in F_p and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in F_p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2022classlacunary,
  author = {Dutykh, D. and Verger-Gaugry, J.-L.},
  title = {On a class of lacunary almost Newman polynomials modulo p and density theorems},
  journal = {Uniform Distribution Theory},
  year = {2022},
  volume = {17},
  number = {1},
  pages = {29--54},
  doi = {10.2478/udt-2022-0007},
  abstract = {The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζ_β (z) of the β-shift, for β > 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in F_p and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in F_p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.},
  keywords = {almost Newman polynomials, Lehmer problem, uniform distribution theory, Salem number, modular representations, Frobenius density theorem, Shusterman, Serre, number of zeroes modulo p, dynamical zeta function, beta-shift, Rosen, modulo p, Stoll, Kronecker's Average Value Theorem, Chebotarev density theorem, lacunary polynomials, density theorems, Schinzel, Sawin, modular forms, Mahler measure, Lacunary integer polynomial, Lehmer's problem, factorization, polynomial zeroes}
}