Finite Exponential Series and Newman Polynomials

Abstract

Abstract.A Newman polynomial is a sum of powers of z, with constant term 1. The Newman polynomial of four terms whose minimum modulus on the unit circle is as large as possible is found by examining the expression /(4) = sup inf and determining an extremal system (xx, ... , x4) using a technique that reduces the problem to a finite search.

On a function related to Chowla's cosine problem

“…The conjectured values of λ(4), λ(5), and λ(6) were obtained by the current author by considering a n ≤ 20. It was conjectured in [5] that µ(n) is monotone, but note that if our conjectured values for λ(5) and λ(6 2 Results…”

“…We note that one can make plausible guesses about other values of λ(n) and µ(n) by searching cosine polynomials or Newman polynomials of bounded degree. The conjectured values of µ(5) and µ(6) given below appear in [5] and were obtained by considering a n ≤ 30. (There is a small error in [5]; the author mistakenly writes the square of the conjectured value of µ (6).…”

“…The conjectured values of µ(5) and µ(6) given below appear in [5] and were obtained by considering a n ≤ 30. (There is a small error in [5]; the author mistakenly writes the square of the conjectured value of µ (6). )…”

Finite searches, Chowla's cosine problem, and large Newman polynomials

“…Campbell et al [2] proved that µ(3) = M(0, 1, 3), and Goddard [6] proved that µ(4) = M(0, 1, 2, 4). In this article, we prove that λ(2) = −L(1, 2) and λ(3) = −L(1, 2, 3), and that 1 ≤ µ(5) ≤ 1 + π/5.…”

Heights of powers of Newman and Littlewood polynomials

“…It is shown in [6] that f (3) is attained at the polynomial 1 + x 2 + x 3 . It seems that this is a mere coincidence, because the polynomial 1 + x 2 + x 3 + x 4 maximizing f (4) (see [13]) is of no use in the study of "small" Q m (P ).…”