On a problem of Byrnes concerning polynomials with restricted coefficients, II

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that rather than . A polynomial with was exhibited for , but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .

     

On a problem regarding coefficients of cyclotomic polynomials

Let an(k) be the coefficient of tk in the nth cyclotomic polynomial

     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

On a discrete norm of the self-inversive unimodular polynomials

The quotient of divided by , whereP is a self-inversive and unimodular polynomial of any degree, dominates an absolute constantK>1. A 1989 paper gaveK=1.0252 on which its authors conjetured that the best constant is . We supply counter examples to their claim and provide a partial result for whenever theL _q norm is replaced by some “discrete” type norm. Research supported by the Shiraz University Grant 72-SC-784-432.     

Littlewood polynomials with high order zeros

Let be the minimal length of a polynomial with coefficients divisible by . Byrnes noted that for each , and asked whether in fact . Boyd showed that for all , but . He further showed that , and that is one of the 5 numbers , or . Here we prove that . Similarly, let be the maximal power of dividing some polynomial of degree with coefficients. Boyd was able to find for . In this paper we determine for .

     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

On roots of polynomials with positive coefficients

We prove that an algebraic number α is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a nonnegative real number. This settles a recent conjecture of Kuba.     

On two theorems of Hardy and Littlewood concerning moment constants

В настоящей работе мы обобщаем два классич еских неравенства Харди и Литтлвуда о моментны х константах. Эти обоб щения получены с использов анием идеи Мателевича и Пав ловича, относящейся к степенным рядам. В новых неравен ствах вместоx ^α иx ^p участвую т общие функцииα(x) и Ф(x).     

On a problem about face polynomials

It is proved that an R-automorphism of polynomial ring R[x₁,…,x_n] is completely determined by its face polynomials, where R is a reduced commutative ring and n≥2. An example is given which shows that the condition R being reduced cannot be weakened.     

On a problem concerning ordered colourings

Let G be a connected graph with v(G) 2 vertices and independence number (G). G is critical if for any edge e of G:

1. (i) (G − e) > (G), if e is not a cut edge of G, and

2. (ii) v(G_i) − (G_i) < v(G) − (G), I = 1, 2, if e is a cut edge and G₁, G₂ are the two components of G − e.

Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then with equality possible if and only if G is a tree. In this paper we establish this conjecture.