A Note on Singularities and Polynomial Zeros
نویسنده
چکیده
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
منابع مشابه
On the polar derivative of a polynomial
For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
متن کاملOn the $s^{th}$ derivative of a polynomial
For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.
متن کاملFurther Observations on Blocking Zeros in Linear Muitivariabie systems (RESEARCH NOTE).
While attempting to clarify the confusion concerning the conceptualization of "blocking zeros" in state space in the recent literature, some new observations are made on the relationship between pole-zero cancellation and transmission blocking. An important distinction between uncontrollable and unobservable eigenvalue s is pointed out; and it is argued that the description of a Blocking Zero, ...
متن کاملPolynomial approximation with doubling weights having finitely many zeros and singularities
We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and not too “rapidly changing” away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generaliz...
متن کاملInequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin
Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$, let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$. Dewan et al proved that if $p(z)$ has all its zeros in $|z| leq k, (kleq 1),$ with $s$-fold zeros at the origin then for every $alphainmathbb{C}$ with $|alpha|geq k$, begin{align*} max_{|z|=...
متن کامل