Source: RJ Lipton website, May 2019
<see the app here: https://duetosymmetry.com/tool/polynomial-roots-toy/>
an app with animation conveying the essence of a mathematical proof? This means more than “proofs in pictures” or “proofs without words”—the animation and interactivity are crucial.
a novel, I think, proof that uses an app. Stein has written the app and it is here. He explains how to use it. I strongly suggest that you try this yourself.
To get a feel for all this, drag the coefficient to and the coefficient to . You should have two real roots in root space (one at , the other at ). Let’s call the negative root, and the positive root. Now move the coefficient around in a small loop (i.e. move it around a little bit, and then return it to where it started). Note that the roots move continuously, and then return to their original positions. Next, move in a big loop (big enough that it orbits around ). Something funny happens: the roots and switch places.