Root Neighborhoods of a Polynomial
Ronald G. Mosier · Mathematics of Computation · 1986
The root neighborhoods of $p(z)$, a polynomial over the complex field, are the sets of complex numbers that are the roots of polynomials which are near to $p(z)$. The term ’near’ means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of $p(z)$. A necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of $p(z)$ and the same number of roots of each near polynomial. Finally, a necessary and sufficient condition is given for a root neighborhood to contain more than one root of $p(z)$, and consequently more than one root of any of the near polynomials.