On Newman and Littlewood polynomials with a prescribed number of zeros inside the unit disk. (English) Zbl 1461.11141
A pair \((k,n)\) \(\in \mathbb{N}^{2},\) where \(1\leq k\leq n-1,\) is said to be
Newman (resp. Littlewood)-admissible, if there is a \(\{0,1\}\) (resp. a \(
\{-1,1\})\)-polynomial of degree \(n,\) having \(k\) zeros with modulus less than
\(1,\) and no zero of modulus \(1.\)
Certain classes of Newman (resp. Littlewood)-admissible pairs have been determined in [K. Mukunda, J. Number Theory 117, No. 1, 106–121 (2006; Zbl 1095.11049); Can. Math. Bull. 53, No. 1, 140–152 (2010; Zbl 1227.11110); P. Borwein et al., Can. J. Math. 67, No. 3, 507–526 (2015; Zbl 1320.11097)], and the authors of the paper under review continue to investigate others sets of these pairs. Mainly, they show that any \((k,n)\) with \(n\geq 7\) and \(3\leq k\leq n-3\) is Newman-admissible. On the way to this goal, they answer a question of D. W. Boyd from [Lond. Math. Soc. Lect. Note Ser. 109, 159–170 (1986; Zbl 0593.12001)] about the smallest degree \(\{0,1\}\)-polynomial having a modulus greater than \(2\) on the unit circle. Also, they identify all pairs \((k,n)\) which are Newman-admissible for \(n\leq 35,\) and conjecture that \((2,n)\) is not Newman-admissible if and only if \(n\equiv 1,7,15,21,25,27\bmod 30\) and \( n\geq 21.\) Similar, but less complete results for Littlewood-admissible pairs are established.
Certain classes of Newman (resp. Littlewood)-admissible pairs have been determined in [K. Mukunda, J. Number Theory 117, No. 1, 106–121 (2006; Zbl 1095.11049); Can. Math. Bull. 53, No. 1, 140–152 (2010; Zbl 1227.11110); P. Borwein et al., Can. J. Math. 67, No. 3, 507–526 (2015; Zbl 1320.11097)], and the authors of the paper under review continue to investigate others sets of these pairs. Mainly, they show that any \((k,n)\) with \(n\geq 7\) and \(3\leq k\leq n-3\) is Newman-admissible. On the way to this goal, they answer a question of D. W. Boyd from [Lond. Math. Soc. Lect. Note Ser. 109, 159–170 (1986; Zbl 0593.12001)] about the smallest degree \(\{0,1\}\)-polynomial having a modulus greater than \(2\) on the unit circle. Also, they identify all pairs \((k,n)\) which are Newman-admissible for \(n\leq 35,\) and conjecture that \((2,n)\) is not Newman-admissible if and only if \(n\equiv 1,7,15,21,25,27\bmod 30\) and \( n\geq 21.\) Similar, but less complete results for Littlewood-admissible pairs are established.
Reviewer: Toufik Zaïmi (Riyadh)
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11R09 | Polynomials (irreducibility, etc.) |
| 12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
| 11C08 | Polynomials in number theory |
| 11B83 | Special sequences and polynomials |
| 65H04 | Numerical computation of roots of polynomial equations |
Keywords:
Newman polynomials; Littlewood polynomials; Pisot numbers; complex Pisot numbers; zero locationCitations:
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