9 - Large Newman polynomials

Summary

INTRODUCTION

Campbell, Ferguson and Forcade [2] have defined a Newman polynomial P(z) to be one with all coefficients either 0 or 1 and with P(0) = 1. Let m(P) denote the minimum of ∣P(z)∣ on the unit circle and let μi(n) be the maximum of m(P) attained for a Newman polynomial of degree n.

The authors [2] exhibited a Newman polynomial P₁₂ of degree 12 with m(P₁₂) > 1. answering a question of D.J. Newman. They computed μ(n) for n ≤ 20 and announced that μ(n) ≤ 1 for n ≤ 11 while μ(n) > 1 for 12 ≤ n ≤ 20 They conjectured that there is a sequence P_n of Newman polynomials for which m(P_n) → ∞, and also that μ(n) > 1 for all n ≥ 12.

The first conjecture was established by Smyth [7], who also showed that μ(n) > 1 for all n ≥ 2194.

In section 2, we give the proof of a strengthening of the first conjecture, namely that μ(n) ≫ n^α for a certain constant α with.137 < α ≤ 1/2.

In the remaining sections, we complete the proof that μ(n) > 1 for all n ≥ 12. Note that Smyth's result reduces this proof to a “finite” amount of computation. since there are only 2²¹⁹³ Newman polynomials of degree less than 2194. One of the main points of our paper is to show how this computation can be reduced to a reasonable size.