SQUEEZING POLYNOMIAL ROOTS A NONUNIFORM DISTANCE

Given a polynomial with all real roots, the Polynomial Root Squeezing Theorem states that moving two roots an equal dis- tance toward each other, without passing other roots, will cause each critical point to move toward (ri + rj)=2, or remain flxed. In this note, we extend the Polynomial Root Squeezing Theorem to the case where two roots are squeezed together a nonuniform distance. p(x) (xiri)(xirj) , and ordinary otherwise. Then the assertion of the Polynomial Root Squeezing Theorem is that if ri and rj move equal distances toward each other, without passing other roots, then each stubborn critical point which is not located at ri or rj will stay flxed, and each ordinary critical point moves toward (ri +rj)=2. If ri or rj is a repeated root of multiplicity greater than two, one of the repeated critical points will move toward (ri + rj)=2, while the others will remain flxed. In this case, the moving root which is closest to a given critical point has the most pull on that critical point. Unfortunately, this intuition does not allow us to see what happens when two distinct roots are squeezed together a nonuniform distance. Throughout the paper we will let p(x) be a polynomial of degree n with n real roots r1 • r2 • ¢¢¢ • rn and critical points c1 • c2 • ¢¢¢ • cni1. Consider two distinct roots ri < rj and ck any ordinary critical point. If we drag ri to the right, the Polynomial Root Dragging Theorem tells us that ck will also move to the right. If we then drag rj to the left, the critical