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Mathematics of Computation

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Root neighborhoods of a polynomial

Author: Ronald G. Mosier
Journal: Math. Comp. 47 (1986), 265-273
MSC: Primary 65G05; Secondary 12D10, 30C10, 30C15
MathSciNet review: 842134
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Abstract: 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.

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Article copyright: © Copyright 1986 American Mathematical Society

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