That is “app” as in an on-line application

 [ Leo Stein ]

Leo Stein is an assistant professor in the department of Physics and Astronomy at the University of Mississippi. His research interests include general relativity from an astrophysical standpoint.

Today I want to share an unusual proof of his.

Mathematics and complexity theory are all about proving theorems. Most of the time, so far, we prove the old way: we write out a humanly readable proof. At least we hope the proof is readable. Some of the time, we use a computer to check or even create the proof. Sometimes we do extensive numerical computations, but these are not proofs.

I have known, as I am sure you do, forever that a quadratic equation can be solved in closed form. That is

$\displaystyle x^{2} + bx + c = 0,$

has the two solutions

$\displaystyle -b/2 + 1/2\sqrt{b^{2}-4c} \text{ and } -b/2 - 1/2\sqrt{b^{2}-4c}.$

I have discussed this before here and its relationship to the World’s Fair in Flushing Meadows.

A natural question is: Are square roots needed in any formula for quadratic equations? The answer is “Yes”.

Theorem 1 There does not exist any continuous function from the space of quadratic polynomials to complex numbers which associates to any quadratic polynomial a root of that polynomial.

Corollary 2 There is no quadratic formula built out of a finite combination of field operations and the functions ${\sin, \cos, \exp}$, and the coefficients of the polynomial.

The corollary uses the basic fact that ${\sin, \cos, \exp}$ are continuous functions. Note that each has a single branch on complex plane, whereas radicals and the logarithm function do not. So how do we prove the theorem?

## An App Based Proof

Here is 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 ${a_{0}}$ coefficient to ${-1}$ and the ${a_{1}}$ coefficient to ${1/2}$. You should have two real roots in root space (one at ${\approx -1.28}$, the other at ${\approx 0.78}$). Let’s call ${r_{1}}$ the negative root, and ${r_{2}}$ the positive root. Now move the coefficient ${a_{0}}$ around in a small loop (i.e. move it around a little bit, and then return it to ${-1}$ where it started). Note that the roots move continuously, and then return to their original positions. Next, move ${a_{0}}$ in a big loop (big enough that it orbits around ${r_{2}}$). Something funny happens: the roots ${r_{1}}$ and ${r_{2}}$ switch places.

Leo Goldmakher says here:

Here is one immediate consequence of this observation:

Theorem 3 There does not exist any continuous function from the space of quadratic polynomials to complex numbers which associates to any quadratic polynomial a root of that polynomial.

And so the corollary follows.

## A Standard Proof

Goldmakher writes out a more conventional proof in his paper titled Arnold’s Elementary Proof Of The Insolvability Of The Quintic. He also shows the following theorem:

Theorem 4 Fix a positive integer ${N}$. Any quintic formula built out of the field operations, continuous functions, and radicals must have nesting of level more than ${N}$.

This says that there can be no fixed formula for fifth degree, quintic, polynomials. Of course, this follows from Galois theory, but his proof uses just calculus. The Arnold is Vladimir Arnold.

## Open Problems

Do you know other cases of 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.

May 18, 2019 3:50 pm

Does a counting frame, as used in elementary school to teach addition, qualify?

May 19, 2019 8:22 pm

Dear Abigail:

I would think that does count…oops bad pun? Did you use one in school?

Best

May 20, 2019 9:18 am

I don’t remember whether we used one — it’s been more than 50 years since I first went to elementary school.

2. May 18, 2019 7:00 pm

Here’s a few Animated Proofs of theorems in propositional calculus, using a variant of C.S. Peirce’s logical graphs. There’s no need for user interaction in these very simple, purely routine proofs, but it would be a good project for someone to supplement the underlying algorithm with user guessing and guidance in more complicated problems.

May 19, 2019 8:21 pm

Dear Jon Awbrey:

I like these animations too. There is the theory that when reading math you should create your own pictures. This helps you learn and understand them. Terry Tao has said this many times. His brilliant books rarely have pictures.

I do like pictures. Thanks for your examples.

Best

• May 23, 2019 7:04 am

Dear Dr Lipton,

May 19, 2019 1:49 am

Here is a piece with a couple of interactive gadgets conveying the essence of an interesting calculus problem: http://hardmath123.github.io/envelope.html

May 19, 2019 6:22 pm

This is beyond superb!

Galois Theory brought alive.

Bravo!!!

May 19, 2019 8:17 pm

Dear Math Boy:

Very nice comment. Glad you like this. I really enjoyed the app too.

Best

6. May 20, 2019 6:03 am

“Next, move {a_{0}} in a big loop (big enough that it orbits around {r_{2}}). Something funny happens: the roots {r_{1}} and {r_{2}} switch places.”

Presumably that should be “… orbits around {a_{1}} …”.

But actually, that’s neither sufficient nor necessary. The critical item is to orbit around the origin; then the roots swap places. You can even drag a_0 in a closed path that encloses a_1 and the roots don’t swap places as long as you avoid enclosing the origin.

May 20, 2019 11:22 am

Hi, Delta. I don’t think it’s the origin that’s the important point. Rather, we need to study the discriminant D of the polynomial P. The discriminant is a polynomial in n variables: the coefficients a_0 … a_{n-1} (since we’ve set a_n=1). Let’s pick one particular a_i and fix all the others, so we can treat the discriminant D(a_i) as a polynomial of at most degree n in the single variable a_i. The roots of the discriminant tell us points in a_i’s complex plane that will lead to P having degenerate roots. Every loop around a root of D generates a transposition of two roots. If you loop around more roots of D, you can generate more complicated permutations.

May 30, 2019 4:46 pm

(longtime reader, first time posting!) This is a great app! Some related links you might enjoy:

1) Frank Sottile and his collaborators have done very interesting work pushing the limits of the numerical approach to computing Galois groups in their work on numeric homotopy continuation. A fun example is the following paper, where they show the Galois group of a natural Schubert problem is the symmetric group on 6006 elements! https://www.math.tamu.edu/~frank.sottile/research/abstracts/Galois.html
This one looks fun too: https://www.math.tamu.edu/~frank.sottile/research/pdf/NGalois.pdf

2) Not knowing about this great app, over the past few years I’ve developed a similar Mathematica script to use for talks to high school students; you might like it for computing more complicated galois groups, as it has the advantage of numbering roots and drawing pretty colored paths between them [at the price of inconvenience and installing Mathematica!] – web.stanford.edu/~seanpkh/rootloops