**Juggling roots**

As you change the coefficients of a polynomial, its roots move around. This is a surprisingly good source of fun for mathematicians of all ages.

This animation by

**twocubes**shows the five roots of

x⁵ + tx³ + 1

moving around as the number t travels around a circle of radius 2 centered at the origin in the complex plane. See more here:

http://curiosamathematica.tumblr.com/post/140731259824/animation-by-twocubes-showing-the-roots-of-the

I think the contours lines are curves where the absolute value of this function is constant, with darker shades where it's smaller, and black where it's zero.

A lot of interesting things in math happen when you have two ways of viewing the same situation: then you can ask how a change in one view corresponds to a change in the other view. The two main ways to view a polynomial are its coefficients and its roots. I can imagine a program where these two views are side by side. You can move the coefficients around in the left side and see how the roots move around at right, or vice versa. For a polynomial of degree 5, dragging one coefficient around a circle in the left-hand view will create this animated image in the right-hand view.

(Be careful: the coefficients are an

*ordered list*of numbers, while the roots are a

*multiset*. Also, to get the roots of a polynomial to determine its coefficients, we should assume it's

**monic**, meaning the first coefficient equals 1. Otherwise you can double all the coefficients without changing the roots.)

Thanks to +Henry Segerman for pointing this out. If any of you haven't seen his posts, it's time to check them out!

#geometry

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- +John Baez wow, the non-conservative stuff! We gave up on trying to get a journal to publish it. Knowing that you (and others) are interested is somewhat validating :)Dec 15, 2017
- +Leo Stein - couldn't publish it??? That's absurd, unless there's a mistake in it or it's been done already. The topic is very interesting, and not just to me. Maybe let's talk privately about how to get that thing published. Maybe you were just trying the wrong journals. Maybe you ran into referees who were convinced it's impossible to have a Lagrangian approach to dissipative systems.Dec 15, 2017
- +John Baez We tried PRX, because we thought it would be interesting to multiple disciplines. Their editors suggested that it was too pedagogical for their journal. We then tried PRD, and got unlucky with a referee. We all moved on to various other things. We
**did**manage to publish work on a numerical integration scheme based on this approach (https://arxiv.org/abs/1506.08443). Dave and Chad each still have a finger in this work in different ways and have published work based on it.

Feel free to email me, DM on twitter, Skype, hangouts, etc.Dec 15, 2017 - +Leo Stein - your paper is too long and well-written for the journals you mentioned. There have got to be some less flashy but still reputable journals you can publish this in. And it deserves to be published, from what I can tell so far! (Or, perhaps split in two and then published: first a paper on the basic concepts, maybe in
*Jour. Math. Phys.*, and then a paper on their application to various dissipative field theories.)

I guess I won't contact you privately (now) because none of this is very touchy. If I make progress in my thinking about dissipative systems I may pester you again!Dec 15, 2017 - +John Baez at the very least we should update the arXiv version, because we spotted some typos during refereeing!

Jour. Math. Phys. sounds like a nice idea. I'll poke Chad and Dave and see if they want to try again :)Dec 15, 2017 - +Leo Stein - I have a theory that young scientists are less persistent than old ones when it comes to publishing what they write. They don't always realize that getting slapped down a few times is par for the course... especially for any work that's unconventional. You guys clearly put a huge amount of work into that paper - so it would be insane (in my humble opinion) not to publish it and add it to your CV.Dec 15, 2017

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