On a problem due to Littlewood concerning polynomials with unimodular coefficients

Littlewood raised the question of how slowly $\lVert f_{n}\rVert_{4}^{4}-\lVert f_{n}\rVert_{2}^{4}$ (where $\lVert.\rVert _{r}$ denotes the L ^r norm on the unit circle) can grow for a sequence of polynomials f _n with unimodular coefficients and increasing degree. The results of this paper are the following. For $$g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k $$ the limit of $(\lVert g_{n}\rVert_{4}^{4}-\lVert g_{n}\rVert_{2}^{4})/\lVert g_{n}\rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=\sum_{j=0}^{n-1}\sum _{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} $$ the limit of $(\lVert h_{n}\rVert_{4}^{4}-\lVert h_{n}\rVert_{2}^{4})/\lVert h_{n}\rVert_{2}^{3}$ is shown to be 4/π ². No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists.     

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 .

     

A remark on two famous theorems concerning polynomials

A relation between Gauss-Lucas Theorem and Laguerre Theorem concerning the zeros of a polynomial in complex domain is discussed.     

The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices

A problem concerning infinite graphs

We show that there are graphs G and H which satisfy: (I) for every integer n, H contains n disjoint graphs each isomorphic to G, and (II) H does not contain infinitely many disjoint graphs each isomorphic to G. This answers one of the questions raised by Halin in the Graph Theory Newsletter.     

A diophantine problem concerning cubes

This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2(n+1) integers, in parametric terms, such that 5n+1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n+2 of the pairwise sums being cubes.     

A note on computing matrix geometric means

A new definition is introduced for the matrix geometric mean of a set of k positive definite n×n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n ³ k ²) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et al. (Linear Algebra Appl 385:305?C334, 2004) have complexity O(n ³ k!2 ^k ). The new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the ten properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.     

Some results concerning growth of polynomials

In this paper, we establish inequalities for polynomials with restricted zeros, which in particular yields interesting generalizations of some Zygmund type inequalities for polynomial.     

Inequalities concerning the derivative of polynomials

A simple proof of Bernstein's Theorem on the derivative of polynomials is presented.     

A problem on the turning points of polynomials

The construction of polynomials with maximum turning points lower than minimum turning points is investigated. As well as providing cautionary examples for students, the solution requires the application of quadratic inequalities and could provide an interesting topic for coursework at A level. The problem also provides an opportunity for practice in the use of graph plotting software.