Newman Polynomials, Reducibility, and Roots on the Unit Circle
-
Idris Mercer
Abstract.
A length k Newman polynomial is any polynomial
of the form 
(where 
). Some Newman polynomials are reducible over the rationals,
and some are not. Some Newman polynomials have roots on the unit circle,
and some do not.
Defining, in a natural way, what we mean by the “proportion”
of length k Newman polynomials with a given property,
we prove that
of length 3 Newman polynomials are reducible over the rationals,
of length 3 Newman polynomials have roots on the unit circle,
of length 4 Newman polynomials are reducible over the rationals,
of length 4 Newman polynomials have roots on the unit circle.
We also show that certain plausible conjectures imply
that the proportion of length 5 Newman polynomials
with roots on the unit circle is 
.
© 2012 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- On Weakly Complete Sequences Formed by the Greedy Algorithm
- Bijective Proofs of Vajda's Ninetieth Fibonacci Number Identity and Related Identities
- Newman Polynomials, Reducibility, and Roots on the Unit Circle
- A Property of Twin Primes
- Number of Permutations with Prescribed Up-Down Structure as a Function of Two Variables
- A -Expansion Associated to Sturmian Sequences
- On the Number of Carries Occurring in an Addition Mod
- A Probabilistic Look at Series Involving Euler's Totient Function
- Fibonacci Variations of a Conjecture of Polignac
- Extending Nathanson Heights to Arbitrary Finite Fields
- On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group
- An Explicit Bound for Aliquot Cycles of Repdigits
- A Note on the Minimal Number of Representations in
- A Unified Proof of Two Classical Theorems on CNS Polynomials
- On the Problem of Molluzzo for the Modulus 4
- On Multiplicative Functions with Bounded Partial Sums
- Avoiding Type or Patterns in a Partition of a Set
Articles in the same Issue
- Masthead
- On Weakly Complete Sequences Formed by the Greedy Algorithm
- Bijective Proofs of Vajda's Ninetieth Fibonacci Number Identity and Related Identities
- Newman Polynomials, Reducibility, and Roots on the Unit Circle
- A Property of Twin Primes
- Number of Permutations with Prescribed Up-Down Structure as a Function of Two Variables
- A -Expansion Associated to Sturmian Sequences
- On the Number of Carries Occurring in an Addition Mod
- A Probabilistic Look at Series Involving Euler's Totient Function
- Fibonacci Variations of a Conjecture of Polignac
- Extending Nathanson Heights to Arbitrary Finite Fields
- On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group
- An Explicit Bound for Aliquot Cycles of Repdigits
- A Note on the Minimal Number of Representations in
- A Unified Proof of Two Classical Theorems on CNS Polynomials
- On the Problem of Molluzzo for the Modulus 4
- On Multiplicative Functions with Bounded Partial Sums
- Avoiding Type or Patterns in a Partition of a Set
