On the zeros of polynomials
The classical Cauchy result [Exercises de mathématiques, IV Année de Bure Fréses, 1829] said that if \[ P(z)= z^n+ a_{n-1}z^{n-1}+\cdots+ a_1z+a_0 \] than all the zeros of \(P(z)\) lie in a circle \(|z|\leq 1+M\) where \(M=\max|a_j|\), \(j=0,1,2,\dots,n-1\). In this paper, the authors extend the above Cauchy result on the class of lacunary type polynomials: Theorem. If \[ P(z)= z^n+ a_pz^p+\cdots+ a_1z+a_0,\quad 0\leq p\leq n-1 \] then all the zeros of \(P(z)\) lie in \(|z|<K\), where \(K\) is a unique positive root of the trinomial \[ X^{n-p}- X^{n-p-1}-M=0 \] for \(M=\max|a_j|\), \(j=0,1,\dots,p\). For \(p=n-1\) this gives the Cauchy result. The other results giving some ring shaped region for the zeros of lacunary polynomials are also present.
scientific article; zbMATH DE number 955215