Matthew Boelkins

Polynomial Root Squeezing

Given a real polynomial with all its roots real, the Polynomial Root Dragging Theorem [1], [2] states that if one or more roots of the polynomial are moved to the right, then all of the critical numbers also move to the right (or possibly stay fixed, if a root is repeated) 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? Anderson’s proof of the Root Dragging Theorem in [2] can be modified to show that for odd and even polynomials, if we drag some subset of the positive roots toward the origin, and simultaneously drag the corresponding negative roots toward the origin so that the updated polynomial remains odd or even, then all of the function’s critical numbers move toward the origin (or stay fixed). For a polynomial that is not necessarily odd or even, the situation is more complicated. As shown in the example in Figure 1, it appears that when two consecutive roots are dragged toward each other (as indicated by the arrows), the critical numbers to the left of the first root move right, while the critical numbers to the right of the second root move left. In particular, for the critical numbers ci of the original polynomial p and the critical numbers di of the updated polynomial q, we have that c1 < d1 and c2 < d2, while c4 > d4. It turns out that our observations in this example hold in general. In what follows we prove the Polynomial Root Squeezing Theorem, which shows that dragging certain pairs of roots toward

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