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Polynomial Root Squeezing

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Abstract:

Given a real polynomial with distinct real zeros, the Polynomial Root Dragging Theorem states that if one or more zeros of the polynomial are moved to the right, then all of the critical numbers also move to the right with none of the critical numbers moving as much as the root that is moved most. But what happens if some of the roots of the polynomial are dragged in opposing directions, either toward or away from each other? The Polynomial Root Squeezing Theorem shows that when two zeros of a polynomial are squeezed together, the outermost critical numbers move inward. We then apply the Root Squeezing Theorem to prove results about which polynomials have derivatives with minimum span; that is, the distance from their derivatives' smallest to greatest zeros is the least possible.

Document Type: Research Article

Publication date: February 1, 2008

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