On the norms of polynomials of two adjoint projections and a shift

LetP be a projection (non-selfadjoint in general), andV a selfadjoint involution acting in a Hilbert spaceH. In this paper the polynomialsF(X, Y, Z) of three non-commuting variables are described such that the norms F(P, P ^*,V) depend only on P. A method of calculation of the norms F(P, P ^*,V) for such polynomials is given. For polynomialsF(P, P ^*) this problem was investigated in [KMF], [FKM].     

关于几类图族伴随多项式的第四项系数

主要研究了几类图族伴随多项式第四项系数的规律，此结果有助于进一步讨论这些图族补图的色唯一性、色等价划分．     

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.     

Minimizing and maximizing the Euclidean norm of the product of two polynomials

We consider the problem of minimizing or maximizing the quotient $$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$ where $p=p_0+p_1x+\dots+p_mx^m$ , $q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]$ , ${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$ , are non-zero real or complex polynomials of maximum degree $m,n\in{\mathbb N}$ respectively and $\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}$ is simply the Euclidean norm of the polynomial coefficients. Clearly f _m,n is bounded and assumes its maximum and minimum values min f _m,n ?=?f _m,n (p _min, q _min) and max f _m,n ?=?f(p _max, q _max). We prove that minimizers p _min, q _min for ${\mathbb K}={\mathbb C}$ and maximizers p _max, q _max for arbitrary ${\mathbb K}$ fulfill $\deg(p_{\min})=m=\deg(p_{\max})$ , $\deg(q_{\min})=n=\deg(q_{\max})$ and all roots of p _min, q _min, p _max, q _max have modulus one and are simple. For ${\mathbb K}={\mathbb R}$ we can only prove the existence of minimizers p _min, q _min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f _m,n for real polynomials which are slightly better than the known ones and inclusions for max f _m,n .     

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.     

Quadratic polynomials represented by norm forms

Let ${P(t) \in \mathbb{Q}[t]}$ be an irreducible quadratic polynomial and suppose that K is a quartic extension of ${\mathbb{Q}}$ containing the roots of P(t). Let ${{\bf N}_{K/\mathbb{Q}}({\rm x})}$ be a full norm form for the extension ${K/\mathbb{Q}}$ . We show that the variety $$\begin{array}{ll}P(t)={\bf N}_{K/\mathbb{Q}}({\rm x})\neq 0\end{array}$$ satisfies the Hasse principle and weak approximation. The proof uses analytic methods.     

On asymptotics of the uniform norm of polynomials with zeros at the roots of unity

The work is related to the problem by P. Erds about the estimation of the numbers

On the existence of the best discrete approximation in norm by reciprocals of real polynomials

For the given data (w_i,x_i,y_i), i=1,…,M, we consider the problem of existence of the best discrete approximation in l_p norm (1≤p<∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l_p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.     

The average norm of polynomials of fixed height

Let be any integer and let

be the set of all polynomials of height 1 and degree . Let

Here is the power of the norm on the boundary of the unit disc. So is the average of the power of the norm over

In this paper we give exact formulae for for various values of . We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.

Theorem 0.1. For , we have

and

     

The expected norm of random polynomials

The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial

let

Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then

and

as .

In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives

and

This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.

     

The norm of minimal polynomials on several intervals

Using works of Franz Peherstorfer, we examine how close the $n$th Chebyshev number for a set $E$ of finitely many intervals can get to the theoretical lower limit $2\mathrm{cap}{\left(E\right)}^n$.     

The value of polynomials represented by norm forms

Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for any smooth proper model of the variety over Q defined by P(t) = NK/Q(x).     

一类图的伴随多项式的导数与积分

设Ｇ是不含三角形的图,本文给出了Ｇ的伴随多项式ｈ（Ｇ,ｘ）的导数和两个积分公式。     

On antipodal and adjoint pairs of points for two convex bodies

The numbers of antipodal and of adjoint pairs of points are estimated for a given pair of disjoint convex bodies inE ^d .     

The norm and the Evaluation of the Macdonald polynomials in superspace

We demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.     

On the Malgrange condition for complex polynomials of two variables

 We show that if P  ℂ[X, Y] is nonconstant, the set K _∞ (P) of points where the Malgrange condition fails to hold coincides with the roots of a constructible polynomial in one variable. This strengthens weaker related results in the literature. The proof given here is purely analytical and bypasses the algebraic and topological arguments involved in other approaches. Applications to both the real and complex Jacobian Conjectures and to the reducibility of P-z are discussed. Received: 11 October 2001 / Revised version: 2 April 2002 Mathematics Subject Classification (2000): 12D05, 14D06, 14R15, 32B10