Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.
An n × n ray pattern A is called a spectrally arbitrary ray pattern if the complex matrices in Q(A) give rise to all possible complex polynomials of degree n. In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an n×n irreducible spectrally arbitrary ray pattern is 3n-1. In this paper, we introduce a new family of spectrally arbitrary ray patterns of order n with exactly 3n - 1 nonzeros., Yinzhen Mei, Yubin Gao, Yanling Shao, Peng Wang., and Obsahuje seznam literatury
We deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the Q-transform employed previously by Varshamov and Garakov (1969) as well as by Meyn (1990) for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper these conditions are removed. We construct infinite sequences of irreducible polynomials of nondecreasing degree starting from any irreducible polynomial., Simone Ugolini., and Obsahuje seznam literatury
A method is given for constructing the proper spline. The necessary expressions for the variances of the normal places taken at two neighbouring polynomials and their covariance are found and given.
Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $\|p'\|_{[-1,1]}\leq\frac12\|p\|_{[-1,1]}$ for a constrained polynomial $p$ of degree at most $n$, initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(-1,1)$ and establish a new asymptotically sharp inequality., Lai-Yi Zhu, Da-Peng Zhou., and Obsahuje bibliografii